Discrete and Continuous Nonlinear Schrodinger systems

Discrete and Continuous Nonlinear Schr¨odinger systems M.J.Ablowitz, B.Prinari and A.D.Trubatch Department of Applied Mathematics, University of Colorado, Campus Box 526, Boulder C0 80309-0526, USA October 18, 2001

Contents 1 Introduction

3

2 Scalar Nonlinear Schr¨ odinger equation 2.1 The inverse scattering transform for NLS . . . . 2.1.1 Direct scattering problem . . . . . . . . . 2.1.2 Inverse Scattering . . . . . . . . . . . . . 2.1.3 Time evolution . . . . . . . . . . . . . . . 2.2 Soliton Solutions . . . . . . . . . . . . . . . . . . 2.3 Conserved Quantities and Hamiltonian structure

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7 7 8 16 20 21 23

3 Integrable Discrete Nonlinear Schr¨ odinger equation 3.1 The Inverse Scattering Transform for IDNLS . . . . . 3.1.1 Overview . . . . . . . . . . . . . . . . . . . . . 3.1.2 Direct scattering problem . . . . . . . . . . . . 3.1.3 Inverse Scattering . . . . . . . . . . . . . . . . 3.1.4 Time evolution . . . . . . . . . . . . . . . . . . 3.2 Soliton solutions . . . . . . . . . . . . . . . . . . . . . 3.3 Conserved quantities and Hamiltonian structure . . .

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27 28 28 29 44 52 54 57

4 Vector Nonlinear Schr¨ odinger equation 4.1 The inverse scattering transform for VNLS . . . . . . . . . . . . 4.1.1 Direct scattering problem . . . . . . . . . . . . . . . . . . 4.1.2 Inverse Scattering . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Time evolution . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Soliton Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 One soliton solution . . . . . . . . . . . . . . . . . . . . . 4.2.2 Transmission coefficients for the pure one soliton potential 4.2.3 Vector Symmetry . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Vector Soliton Interactions . . . . . . . . . . . . . . . . . 4.3 Conserved quantities and Hamiltonian structure . . . . . . . . .

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59 59 60 69 73 74 74 75 75 76 79

5 Discrete matrix NLS 5.1 Overview . . . . . . . . . . . . . . 5.2 The Inverse Scattering Transform . 5.2.1 Direct Scattering Problem . 5.2.2 Inverse Scattering Problem 5.2.3 Time evolution . . . . . . . 5.3 Vector Solitons . . . . . . . . . . .

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82 . 82 . 84 . 84 . 105 . 121 . 122

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5.4

5.3.1 One soliton solutions . . . . 5.3.2 Transmission coefficients for 5.3.3 Vector soliton interactions . Conserved Quantities . . . . . . . .

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122 123 126 131

A Summation by Parts Formula

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B Transmission of the Jost function through a localized potential

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C Scattering theory for the discrete Schr¨ odinger equation C.1 Direct Scattering Problem . . . . . . . . . . . . . . . . . . C.1.1 Existence and analyticity of the Jost functions . . C.1.2 Scattering Data . . . . . . . . . . . . . . . . . . . . C.1.3 Symmetries . . . . . . . . . . . . . . . . . . . . . . C.1.4 Eigenvalues and norming constants . . . . . . . . . C.2 Inverse Problem . . . . . . . . . . . . . . . . . . . . . . . . C.2.1 Recovery of the Jost functions . . . . . . . . . . . C.2.2 Recovery of the potential . . . . . . . . . . . . . .

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138 139 141 143 144 146 147 147 150

Chapter 1

Introduction This paper is devoted to the study of integrable discrete and continuous nonlinear Schr¨odinger systems. The purpose is to put most of the known results, and some new ones, into a comprehensive and unified framework within the Inverse Scattering Transform approach (IST). The IST method used to solve both discrete and continuous nonlinear equations is well known; the reader can find numerous monographs describing the mathematical formulation (cf. for instance [1], [2]). In this paper we essentially follow the methodology discussed in the monograph [3]. The analysis involves 1. Direct Problem: (a) obtaining analytic eigenfunctions using Green’s function methods (b) finding relevant scattering data (c) finding the symmetries of the data 2. Inverse Problem: (a) formulating a generalized Riemann-Hilbert (RH) boundary value problem which takes into account the analytic properties of the eigenfunctions (b) obtaining Gel’fand-Levitan-Marchenko integral equations (c) obtaining reflectionless potentials from the RH problem 3. Time Dependence: (a) finding the time evolution of the scattering data (b) obtaining explicit soliton solutions (c) for vector NLS systems, obtaining explicit formulae for the polarization shift which is relevant to vector soliton collisions. In 1965 Zabusky and Kruskal [4] discovered that the solitary wave solution to the KdV equation had the property of interacting “elastically” with another such solution and they called these solutions “solitons”. Shortly afterward [5], Gardner, Green, Kruskal and Miura proposed a method for solving the KdV equation by making use of the ideas of direct and inverse scattering. In 1972, Zakharov and Shabat [6] showed that the method indeed worked also for another physically significant nonlinear evolution equation, namely the nonlinear Schr¨odinger equation (NLS). Using 3

these ideas, in [7] Ablowitz, Kaup, Newell and Segur developed a method to find a rather wide class of nonlinear evolution equations solvable by this technique which they named Inverse Scattering Transform (IST), since, in analogy with the Fourier transform for linear equations, it allows one to exactly solve the initial value problem for a nonlinear evolution equation. Shortly thereafter, significant progress was made in the understanding of certain discrete problems. In [8] Flaschka showed how IST could be used to solve the Toda lattice equation and Manakov [9] used a similar formulation to solve a nonlinear ladder network. Ablowitz and Ladik [10]-[11] provided a broad formulation of the IST method for discrete problems and found that integrable semi-discrete and doubly discrete NLS equations were solvable. The above equations are all 1 + 1 (one space - one time) dimensional. We note that in the 1980’s significant progress was made in the study of IST in 2 + 1 dimensions (a review of some of this work can be found in [3]). The Davey-Stewartson equation, for instance, provides a natural 2 + 1 dimensional integrable extension of the NLS equation, but its study is outside the scope of what we will discuss in this paper. The scalar nonlinear Schr¨ odinger equation (NLS) 2

iqt = qxx ± 2 |q| q

(1.1)

is a physically and mathematically significant nonlinear evolution equation. The NLS equation arises in a generic situation. It describes the evolution of small amplitude, slowly varying wave packets in nonlinear media [13]. Indeed, it has been derived in such diverse fields as: deep water waves [15], [14], plasma physics [16], nonlinear optical fibers [17], magnetostatic spin waves [18] etc. Mathematically it attains broad significance since it is integrable by IST, admits soliton solutions, has an infinite number of conserved quantities etc. We also note that the form of the NLS equation (1.1) with a minus sign in front of the nonlinear term is sometimes referred to as the “defocusing” case. The defocusing NLS equation does not admit soliton solutions which vanish at infinity. However, it does admit soliton solutions which have a nontrivial background intensity (called dark solitons). We will only discuss the IST for functions decaying sufficiently rapidly at infinity. The vector nonlinear Schr¨ odinger equation (VNLS) arises, physically, under conditions similar to those described by NLS when there are multiple wavetrains. Also,VNLS models systems where the field has more that one component. For example, in optical fibers and waveguides, the propagating electric field has two components transverse to the direction of propagation. The system   (1) (1) (1.2) + 2 |q (1) |2 + |q (2) |2 q (1) iqt = qxx   (2) (2) iqt = qxx + 2 |q (1) |2 + |q (2) |2 q (2)

as an asymptotic model for the propagation of the electric field in a waveguide was first examined by Manakov [12]. Subsequently, this system was derived as a key model for light-wave propagation in optical fibers [19]–[21]. In literature this equation is sometimes referred to as the coupled NLS equation. Both the VNLS equation (1.2) and its vector generalization 2

iqt = qxx ± 2 kqk q

(1.3)

where q is an N component vector and k·k is the Euclidean norm, are integrable by the IST. In [12] only the case N = 2 is studied, but the extension to more components is straightforward. The N -component equation can be derived, with some additional conditions, as an asymptotic model of the interaction of N wavetrains in a weakly nonlinear, conservative medium. 4

In optical fibers and waveguides, depending on the physics of the particular system, the propagation of the electromagnetic waves may be described by variations of equation (1.2). Note that the VNLS equation is the ideal (exactly integrable) case. For example, a model with physical significance is   (1) (1) (1.4) iqt = qxx + 2 |q (1) |2 + B|q (2) |2 q (1)   (2) (2) iqt = qxx + 2 B|q (1) |2 + |q (2) |2 q (2)

which is equivalent to the eq. (1.2) when B = 1. However, based on the properties of the equation (1.4), apparently it is not integrable when B 6= 1. VNLS (1.3) has a natural matrix generalization in the system iQt = Qxx − 2QRQ

(1.5)

−iRt = Rxx − 2RQR

(1.6)

where Q and R are N × M and M × N matrices, respectively, and the superscript H denotes the Hermitian (conjugate transpose). When R = ∓QH the system (1.5)–(1.6) reduces to the single matrix equation iQt = Qxx ± 2QQH Q

(1.7)

which we refer to as matrix NLS or MNLS. VNLS corresponds to the special case when Q is an N -component row vector and R is an M -component column vector or vice-versa. In particular, we obtain equation (1.2) when N = 2. Both the NLS and the VNLS equations admit integrable discretizations which, besides being used as the basis for constructing numerical schemes for the continuous counterparts, also have physical applications as discrete systems (see e.g. [23]-[26]). A natural discretization of NLS (1.1) is the following i

d 1 qn = 2 (qn+1 − 2qn + qn−1 ) ± |qn |2 (qn+1 + qn−1 ) dt h

(1.8)

which here is referred to as the integrable discrete NLS (IDNLS). It is a O(h2 ) finite-difference approximation of (1.1) which is integrable via the IST and has soliton solutions on the infinite 2 lattice [10]- [11]. We note that if we change the nonlinear term in (1.8) to 2 |qn | qn , the equation in apparently no longer integrable and it has been found that in certain circumstances chaotic dynamics results [30]. Correspondingly, we will consider the discretization of VNLS given by the following system 1 d qn = 2 (qn+1 − 2qn + qn−1 ) − rn · qn (qn+1 + qn−1 ) dt h 1 d −i rn = 2 (rn+1 − 2rn + rn−1 ) − rn · qn (rn+1 + rn−1 ) dt h i

(1.9) (1.10)

where qn and rn are N component vectors and · is the inner product. Under the symmetry reduction rn = ∓q∗n (here and in the following ∗ indicates the complex conjugate), the system (1.9)–(1.10) reduces to the single equation i

d 1 qn = 2 (qn+1 − 2qn + qn−1 ) ± kqn k2 (qn+1 + qn−1 ) dt h

5

(1.11)

which, for qn = q(nh) in the limit h → 0, nh = x gives VNLS (1.3). In [29] it was shown that solitary wave solutions interact elastically, and that (1.11) admits multisoliton solutions. Thus the expectation was that the discrete vector NLS system (1.11) is indeed integrable. An associated pair of linear operators (Lax pair) for the system (1.9)–(1.10) was constructed in [28]. Actually, the system is a reduction of a matrix generalization derived in [29]-[31] of the Lax pair pair used in the IST of IDNLS (1.8). Indeed, the matrix analog of IDNLS (1.8) is given by d Qn = Qn+1 − 2Qn + AQn + Qn B + Qn−1 − Qn+1 Rn Qn − Qn Rn Qn−1 dτ d −i Rn = Rn+1 − 2Rn + BRn + Rn A + Rn−1 − Rn+1 Qn Rn − Rn Qn Rn−1 dτ i

(1.12) (1.13)

where Qn , Rn are N × M and M × N matrices, respectively, A is an N × N diagonal matrix and B is an M × M diagonal matrix. A and B represent a gauge freedom in the definition of integrable discrete MNLS which will be used in the following. Note that the system (1.12)–(1.13) does not, in general, admit the reduction Rn = ∓QH n.

(1.14)

However, for N = M one can restrict Rn and Qn to be such that Rn Qn = Qn Rn = αn I,

(1.15)

where I is the identity N × N matrix and αn is a scalar, and make Rn = ∓QH n a consistent reduction of the system (1.12)–(1.13) resulting in the single matrix equation i

d Qn = Qn+1 − 2Qn + AQn + Qn B + Qn−1 − αn (Qn+1 + Qn−1 ) . dτ

(1.16)

Similarly, the IST follows the same lines as that presented for (1.12)–(1.13) with additional symmetry conditions imposed. The additional symmetry (1.15) (which has no analog in the continuous case) has essential consequences for the IST which will be illustrated in detail. The outline of the paper is the following: in Chap. 2 we review the results for the scalar NLS equation; in Chap. 3 we consider the integrable discretization of NLS equation, eq. (1.8); in Chap. 4 we consider VNLS (1.2) and in Chap. 5 we deal with the discrete matrix NLS (DMNLS), i.e. the system (1.12)–(1.13). In all cases we solve the direct problem and find the explicit regularity conditions on the potentials for which such problem is well-defined, formulate the inverse scattering problem as a Riemann-Hilbert boundary value problem and derive from it Gel’fandLevitan-Marchenko integral equations, find the time evolution and explicit soliton solutions. We also consider the problem of a multisoliton collision and find an analytic formula for the polarization shift of vector solitons (this last result is new as far as the discrete vector solitons are concerned).

6

Chapter 2

Scalar Nonlinear Schr¨ odinger equation 2.1

The inverse scattering transform for NLS

The scalar NLS equation (1.1) is well known to be the compatibility condition of the following linear equations (cf. [1]-[2])   −ik q v (2.1) vx = r ik and vt =

2ik 2 + iqr −2kr + irx

−2kq − iqx −2ik 2 − iqr

!

v

(2.2)


T where v is a 2-component vector, v(x, t) = v (1) (x, t), v (2) (x, t) and q = q(x, t), r = r(x, t). The term compatibility condition arises from the fact that the equality of the mixed derivatives, i.e. vxt = vtx , is equivalent to the statement that q and r satisfy the evolution equations iqt = qxx − 2rq 2 −irt = rxx − 2qr2

(2.3) (2.4)

if k, the scattering parameter, is independent of x and t. The system (2.3)–(2.4) reduces to the single PDE (1.1) under the reduction r = ∓q ∗ . We refer to the equation with the x derivative, eq. (2.1), as the scattering problem and the equation with the t derivative, eq. (2.2), as the time dependence. In this section we develop the inverse scattering transform for the NLS equation, eq. (1.1), on the infinite line. The solution q of NLS, at a fixed time, is the potential of the scattering problem (2.1). We formulate the IST procedure for the somewhat more general system (2.3)–(2.4) and then consider the reductions r = ∓q ∗ as a special case. The IST can be broken into three parts: (i) the direct problem - constructing x-independent scattering data from the potential - (ii) the inverse problem - reconstructing the potential from the scattering data - (iii) time evolution - determining the evolution of the scattering data by making use of the time-dependence operator (2.2). The IST procedure for solving the initial-value problem proceeds by first constructing the scattering data at t = ti from the initial data q(ti ), r(ti ) - i.e. 7

step (i) - then computing the evolution of the scattering data from ti to t 6= ti - step(iii) - and, finally, recovering q(t), r(t) by solving the inverse problem - step (ii). The treatment of the direct problem given here follows [1],[3]. The inverse problem is formulated as a Riemann-Hilbert boundary value problem, following [3]. The Gelfand-Levitan-Marchenko integral equations follow from the RH problem.

2.1.1

Direct scattering problem

Jost functions and integral equations We refer to solutions of the scattering problem (2.1) as eigenfunctions with respect to the parameter k. When the potentials q, r → 0 rapidly as x → ±∞, the eigenfunctions are asymptotic to the solutions of   −ik 0 v vx = 0 ik when |x| is large. Therefore it is natural to introduce the eigenfunctions defined by the following boundary conditions     1 0 −ikx ¯ φ(x, k) ∼ e , φ(x, k) ∼ eikx as x → −∞ (2.5) 0 1   0 ψ(x, k) ∼ eikx , 1

¯ k) ∼ ψ(x,

  1 e−ikx 0

as x → +∞.

(2.6)

In the following analysis, it is convenient to consider functions with constant boundary conditions. Hence, we define the Jost functions as follows ¯ k), ¯ (x, k) = e−ikx φ(x, M ¯ k). ¯ (x, k) = eikx ψ(x, N

M (x, k) = eikx φ(x, k), N (x, k) = e−ikx ψ(x, k),

(2.7)

If the scattering problem (2.1) is rewritten as vx = (ikJ + Q) v

(2.8)

where J=



−1 0 0 1



,

Q=



0q r0



(2.9)

¯ (x, k) are solutions and I denotes the 2 × 2 identity matrix, then the Jost functions M (x, k) and N of the differential equation χx (x, k) = ik(J + I)χ(x, k) + (Qχ) (x, k)

(2.10)

¯ (x, k) satisfies while N (x, k) and M χ ˜x (x, k) = ik(J − I)χ(x, ˜ k) + (Qχ) ˜ (x, k) with the constant boundary conditions   1 M (x, k) ∼ , 0   0 N (x, k) ∼ , 1

  0 1   ¯ (x, k) ∼ 1 N 0 ¯ (x, k) ∼ M

8

(2.11)

as x → −∞

(2.12)

as x → +∞.

(2.13)

Solutions of the differential equations (2.10)–(2.11) can be represented by means of the following integral equations Z +∞ χ(x, k) = w + G(x − ξ, k) (Qχ) (ξ, k)dξ −∞ +∞

χ(x, ˜ k) = w ˜+

Z

˜ − ξ, k) (Qχ) G(x ˜ (ξ, k)dξ

−∞

or, in component form, for j = 1, 2 χj (x, k) = wj + χ ˜j (x, k) = w ˜j +

Z

+∞

2 X

−∞ `=1 Z +∞ X 2 −∞ `=1

Gj` (x − ξ, k) (Qχ)` (ξ, k)dξ ˜ j` (x − ξ, k) (Qχ) G ˜ ` (ξ, k)dξ

T

where w = (w1 , 0) , w ˜ = (0, w ˜2 )T and the (matrix) Green’s functions G(x, k) = (Gj` (x, k))j,`=1,2   ˜ j` (x, k) ˜ satisfy the differential equations and G(x, k) = G j,`=1,2

˜ L+ L− 0 G(x, k) = δ(x)I, 0 G(x, k) = δ(x)I ± L0 = I ∂x − ik(J ± I). The Green’s functions are not unique, and, as we show below, the choice of the Green’s function and the choice of the inhomogeneous term together determine the Jost function and its analytic properties. Using the Fourier transform method, it is easy to find  Z  −1 1 p 0 G(x, k) = eipx dp 0 (p − 2k)−1 2πi C  Z  1 (p + 2k)−1 0 ˜ eipx dp G(x, k) = 0 p−1 2πi C˜ ˜ ± (x, k) defined where C and C˜ are appropriate contours. It is natural to consider G± (x, k) and G by  Z  −1 1 p 0 eipx dp G± (x, k) = 0 (p − 2k)−1 2πi C±  Z  1 (p + 2k)−1 0 ˜ G± (x, k) = eipx dp 0 p−1 2πi C± where C± are the contours from −∞ to +∞ which, respectively, pass below and above both the singularities at p = 0 and p = 2k (see Fig. 2.1). Therefore    −2ikx  1 0 e 0 ˜ G± (x, k) = ±θ(±x) , G (x, k) = ∓θ(∓x) (2.14) ± 0 e2ikx 0 1 where θ(x) is the Heaviside function (θ(x) = 1 if x > 0, θ(x) < 0 if x < 0). The “+” functions are analytic in the upper half plane of k and the “−” functions are analytic in the lower half plane. 9

C+

.

.

p=0

p=2k

C−

Figure 2.1: The contours C+ and C−

Taking into account the boundary conditions (2.12)–(2.13), we get the following integral equations for the Jost solutions   Z +∞ 1 G+ (x − ξ, k) (QM ) (ξ, k)dξ (2.15) + M (x, k) = 0 −∞   Z +∞ 0 ˜ + (x − ξ, k) (QN ) (ξ, k)dξ + G (2.16) N (x, k) = 1 −∞   Z +∞
¯ (ξ, k)dξ ¯ (x, k) = 0 + ˜ − (x − ξ, k) QM G (2.17) M 1 −∞   Z +∞
¯ (x, k) = 1 + ¯ (ξ, k)dξ. N G− (x − ξ, k) QN (2.18) 0 −∞

Eqs. (2.15)–(2.18) are Volterra integral equations and we show in the following Lemma 2.1 that if q,r ∈ L1 (R), the Neumann series of the integral equations for M and N converge absolutely and uniformly (in x and k) in the upper half k-plane, while the Neumann series of the integral ¯ and N ¯ converge absolutely and uniformly (in x and k) in the lower half k-plane. equations for M This fact immediately implies that M (x, k) and N (x, k) are analytic functions of k for Imk > 0 ¯ (x, k), N ¯ (x, k) are analytic functions of k for Imk < 0 and and continuous for Im k ≥ 0 and M continuous for Im k ≤ 0. Lemma 2.1 If q,r ∈ L1 (R) then M (x, k), N (x, k) defined by (2.15)–(2.16) are analytic functions ¯ (x, k), N ¯ (x, k) defined by (2.17)–(2.18) are analytic functions of k for of k for Im k > 0, while M Im k < 0. Proof 2.1 We will prove the result for M (x, k). Analogously it can be proved for the remaining eigenfunctions. The Neumann series M (x, k) =

∞ X

C (j) (x, k)

(2.19)

j=0

where C

(j+1)

(x, k) =

Z

+∞

−∞

  1 (0) C (x, k) = 0

  G+ (x − ξ, k) QC (j) (ξ, k)

(2.20) (2.21)

10

is, formally, a solution of the integral equation (2.15). In component form Z x C (j+1),1 (x, k) = q(ξ)C (j),2 (ξ, k)dξ −∞ Z x (j+1),2 C (x, k) = e2ik(x−ξ) r(ξ)C (j),1 (ξ, k)dξ. −∞

Because C (0),2 = 0 we have for any j ≥ 0 C (2j+1),1 = 0,

C (2j),2 = 0.

Using the identities 1 j!

Z

x

−∞

|f (ξ)|

"Z

ξ

0

|f (ξ )| dξ

−∞

0

#j

1 dξ = (j + 1)! 1 = (j + 1)!

Z

x

−∞

Z

d dξ

"Z

ξ

0

|f (ξ )| dξ

−∞

x

|f (ξ)| dξ

−∞

0

#j+1

dξ

j+1

where f ∈ L1 (R), one can easily show by induction that for Im k ≥ 0 j+1 j R R x x |r(ξ)| dξ |q(ξ)| dξ −∞ −∞ (2j+1),2 (x, k) ≤ C j! (j + 1)! R j R j x x |q(ξ)| dξ |r(ξ)| dξ −∞ −∞ (2j),1 . (x, k) ≤ C j! j!

Therefore, if q, r ∈ L1 (R) the series (2.19) is majorized in norm by a uniformly convergent power series, which proves that the Neumann series (2.19) is itself uniformly convergent for Im k ≥ 0. Simply requiring q, r ∈ L1 (R) does not yield analyticity on the real axis; more stringent conditions must be imposed. For instance, using the ideas in Lemma 2.1, one can show that if |r(x)| ≤ Ce−2K|x| ,

|q(x)| ≤ Ce−2K|x|

where C and K are some positive constants, then M and N are analytic for all k with Im k ≥ −K ¯,N ¯ are analytic for all k with Im k < K. Having r, q vanishing faster than any exponential and M as |x| → ∞ implies that all four eigenfunctions are entire functions of k (Volterra equations on a finite interval always have absolutely convergent Neumann series solutions). From the integral equations (2.15)–(2.18) we may compute the asymptotic expansion for large k of the Jost functions. Integration by parts yields ! Rx 1 1 − 2ik −∞ q(ξ)r(ξ)dξ + O(k −2 ) (2.22) M (x, k) = 1 − 2ik r(x) ! R +∞ 1 q(ξ)r(ξ)dξ 1 + 2ik x ¯ + O(k −2 ) (2.23) N (x, k) = 1 r(x) − 2ik ! 1 2ik q(x) N (x, k) = R +∞ + O(k −2 ) (2.24) 1 1 − 2ik q(ξ)r(ξ)dξ x ! 1 2ik q(x) ¯ Rx M (x, k) = + O(k −2 ). (2.25) 1 1 + 2ik q(ξ)r(ξ)dξ −∞ 11

Scattering Data The two eigenfunctions with fixed boundary conditions as x → −∞ are linearly independent, as are the two eigenfunctions with fixed boundary conditions as x → +∞. Indeed, if u(x, k) =
T u(1) (x, k), u(2) (x, k) and v(x, k) = (v (1) (x, k), v (2) (x, k))T are any two solutions of (2.1), we have d W (u, v) = 0 dx

(2.26)

where the Wronskian of u and v is given by W (u, v) = u(1) (x, k)v (2) (x, k) − u(2) (x, k)v (1) (x, k). From the asymptotics (2.5)-(2.6) it follows

¯ k) = 1 W φ, φ¯ = lim W φ(x, k), φ(x, x→−∞

¯ k) = −1 W ψ, ψ¯ = lim W ψ(x, k), ψ(x, x→+∞

(2.27)

(2.28) (2.29)

¯ Therefore which proves that the functions φ, φ¯ are linearly independent, and so are ψ and ψ. ¯ ¯ we can write φ(x, k) and φ(x, k) as linear combinations of ψ(x, k) and ψ(x, k) or vice-versa. The coefficients of these linear combinations depend on k. Hence, the relations ¯ k) φ(x, k) = b(k)ψ(x, k) + a(k)ψ(x, ¯ k) = a ¯ k) φ(x, ¯(k)ψ(x, k) + ¯b(k)ψ(x,

(2.30) (2.31)

hold for any k such that all four eigenfunctions exist. In particular, (2.30)–(2.31) hold for Im k = 0 and define ¯(k), b(k) and ¯b(k). Comparing the asymptotics of
the scattering coefficients a(k), a ¯ W φ, φ as x → ±∞ with eqs. (2.30)–(2.31) shows that the scattering data satisfy the following characterization equation a(k)¯ a(k) − b(k)¯b(k) = 1.

(2.32)

The scattering coefficients can be represented as Wronskian of the Jost functions. Indeed, from eqs. (2.30)–(2.31) it follows ¯ ψ) ¯ a(k) = W (φ, ψ), a ¯(k) = −W (φ,

¯ ¯ ¯ b(k) = −W φ, ψ , b(k) = W φ, ψ .

(2.33) (2.34)

Therefore, as long as q, r ∈ L1 (R), due to the result of Lemma 2.1, (2.33) immediately imply that a(k) is analytic in the upper k plane while a ¯(k) is analytic in the lower k plane. In general, b and ¯b cannot be extended off the real k-axis. Alternatively, one can derive the following integral relationships for the scattering coefficients Z +∞ a(k) = 1 + q(ξ)M (2) (ξ, k)dξ (2.35) b(k) =

Z

−∞ +∞ −2ikξ

e

r(ξ)M (1) (ξ, k)dξ

(2.36)

−∞

a ¯(k) = 1 + ¯b(k) =

Z

Z

+∞

¯ (1) (ξ, k)dξ r(ξ)M

−∞ +∞ 2ikξ

e

¯ (2) (ξ, k)dξ q(ξ)M

−∞

12

(2.37) (2.38)

¯ (j) for j = 1, 2 denote the j-th component of vectors M and M ¯ respectively. where M (j) , M Indeed, let us introduce ¯ (x, k). ∆(x, k) = M (x, k) − a(k)N ¯ and the relation Using the integral eqs. (2.15), (2.18) for M and N   1 0 G+ (x, k) − G− (x, k) = 0 e2ikx we can write ∆(x, k) −

Z

+∞



1 − a(k) 0

G− (x − ξ, k) (Q∆) (ξ, k)dξ

−∞

=



−

Z

+∞

−∞

q(ξ)M (2) (ξ, k) e2ik(x−ξ) r(ξ)M (1) (ξ, k)

!

dξ.

(2.39)



(2.40)

From the other side, the scattering equation (2.30) yields ∆(x, k) = b(k)e2ikx N (x, k) and then ∆(x, k) −

Z

+∞

G− (x − ξ, k) (Q∆) (ξ, k) dξ =

−∞



0 b(k)e2ikx

where we used the integral equation (2.16) for N (x, k), as well as the identity ˜ + (x, k). G− (x, k) = e2ikx G Comparing (2.39) and (2.40) we get the integral representations (2.35) and (2.36). Eqs. (2.37) and ¯ ¯ (x, k) − a (2.38) are derived analogously, by considering ∆(x, k) = M ¯(k)N (x, k). From the integral representations (2.35) and (2.37) and the asymptotics (2.22), (2.25) it also follows Z +∞ 1 q(ξ)r(ξ)dξ + O(k −2 ) Im k > 0 (2.41) a(k) = 1 − 2ik −∞ Z +∞ 1 a ¯(k) = 1 + q(ξ)r(ξ)dξ + O(k −2 ) Im k < 0. (2.42) 2ik −∞ Note that the eqs.(2.30) and (2.31) can be written as ¯ (x, k) + ρ(k)e2ikx N (x, k) µ(x, k) = N ¯ (x, k) µ ¯(x, k) = N (x, k) + ρ¯(k)e−2ikx N

(2.43) (2.44)

where we introduced µ(x, k) = M (x, k)a−1 (k),

¯ (x, k)¯ µ ¯(x, k) = M a−1 (k)

(2.45)

ρ¯(k) = ¯b(k)¯ a−1 (k).

(2.46)

and the reflection coefficients ρ(k) = b(k)a−1 (k), 13

Proper Eigenvalues and Norming Constants A proper eigenvalue of the scattering problem (2.1) is a complex value of k (Im k 6= 0) corresponding to a bounded solution v such that v → 0 as x → ±∞. Suppose a(kj ) = 0 for some kj = ξj + iκj , κj > 0. Then from (2.33) it follows that W (φ(x, kj ), ψ(x, kj )) = 0 and therefore φj (x) = φ(x, kj ) and ψj (x) = ψ(x, kj ) are linearly dependent, i.e. there exists a complex constant cj such that φj (x) = cj ψj (x).

(2.47)

Hence, by (2.5) and (2.6) it follows   1 φj (x) ∼ eκj x−iξj x 0 φj (x) = cj ψj (x) ∼ cj

as x → −∞

  0 e−κj x+iξj x 1

as x → +∞

and therefore kj is a proper eigenvalue. From the other side, if a(k) 6= 0, then any solution of the scattering problem blows up in one or both directions. Hence, the proper eigenvalues in the region Im k > 0 are the zeros of a(k). Similarly, the eigenvalues in the region Im k < 0 are the zeros of a ¯(k) and these zeros k¯j = ¯ ¯ ξj − i¯ κj , κ ¯ j > 0 for j = 1, . . . , J, are such that φ¯j (x) = c¯j ψ¯j (x)

(2.48)

¯ k¯j ), ψ¯j (x) = ψ(x, ¯ k¯j ). {cj }J for some complex constant c¯j , where, as before, φ¯j (x) = φ(x, j=1  J¯ and c¯j j=1 are called norming constants. In terms of the eigenfunctions with fixed boundary conditions, the norming constants are defined by ¯ j (x) = e−2ik¯j x c¯j N ¯j (x). M

Mj (x) = e2ikj x cj Nj (x),

(2.49)

If the potentials q, r ∈ L1 (R), from the analyticity properties of a and a ¯ it follows that the eigenvalues are in finite number and moreover, due to the fact that a and a ¯ are continuous for Im k ≥ 0 and Im k ≤ 0 respectively, no cluster points of zeros are allowed on the real k axis. We further assume that the eigenvalues are simple zeros of a(k) and a ¯(k). If the eigenvalues are not simple zeros, one can study the situation by the coalescence of simple poles (see [6]). Note that if the potentials are rapidly decaying so that eqs. (2.30)–(2.31) can be analytically extended off the real axis, then cj = b(kj ),

c¯l = ¯b(k¯l ),

j = 1, . . . , J

¯ l = 1, . . . , J.

Symmetry Reductions The evolution equation (1.1) is a special case of the system (2.3)–(2.4) under the symmetry reduction r = ∓q ∗ .

(2.50)

This symmetry in the potential induces a symmetry betwen the Jost functions analytic in the upper half k-plane (UHP) and the Jost functions analytic in the lower half k-plane (LHP). In turn, this symmetry of the Jost functions induces a symmetry in the scattering data. 14

Indeed, if v(x, k) = (v (1) (x, k), v (2) (x, k))T satisfies eq. (2.1) and symmetry (2.50) holds, then also vˆ(x, k) = (v (2) (x, k ∗ ), ∓v (1) (x, k ∗ ))H satisfies the same equation. Taking into account the boundary conditions (2.5)–(2.6), we get ¯ k) = ψ(x, ¯ (x, k) = N





ψ (2) (x, k ∗ ) ∓ψ (1) (x, k ∗ )

∗

N (2) (x, k ∗ ) ∓N (1) (x, k ∗ )

∗ ∓φ(2) (x, k ∗ ) φ(1) (x, k ∗ ) ∗  (2) ∗ ¯ (x, k) = ∓M (x, k ) . M M (1) (x, k ∗ )

¯ k) = φ(x,

,

∗

,



(2.51) (2.52)

Then from the integral representations (2.35)–(2.38) for the scattering data it easily follows that a ¯(k) = a∗ (k ∗ ) ¯b(k) = ∓b∗ (k ∗ )

(2.53) (2.54)

and consequently ρ¯(k) = ∓ρ∗ (k)

Im k = 0.

(2.55)

From (2.53) it follows that kj is a zero of a(k) in the UHP iff kj∗ is a zero for a ¯(k) in the LHP, which yields, due to (2.49), J = J¯ and k¯j = kj∗ ,

c¯j = ∓c∗j

j = 1, . . . , J.

(2.56)

Note that when r = q ∗ , the operator (2.1) is Hermitian. In this case, for q → 0 sufficiently rapidly as |x| → ∞, there are no eigenvalues with Im k 6= 0 (cf. [7]). Trace formula J

Assume a(k) and a ¯(k) to have the simple zeros {kj : Im kj > 0}j=1 and respectively, and define

 J¯ k¯j : Im k¯j < 0 j=1 ,

J¯ ∗ Y k − k¯m α ¯ (k) = a ¯(k). k − k¯m m=1

J ∗ Y k − km a(k), α(k) = k − km m=1

(2.57)

α(k) is analytic in the UHP, whereas it has no zeros, while α ¯ (k) is analytic in the LHP, whereas it has no zeros; moreover, due to (2.41), α(k), α(k) ¯ → 1 as |k| → ∞ in the proper half plane. Therefore we have Z +∞ Z +∞ 1 1 log α(ξ) log α ¯ (ξ) log α(k) = dξ, dξ = 0 Im k > 0 2πi −∞ ξ − k 2πi −∞ ξ − k Z +∞ Z +∞ 1 1 log α ¯ (ξ) log α(ξ) log α(k) ¯ =− dξ, dξ = 0 Im k < 0. 2πi −∞ ξ − k 2πi −∞ ξ − k Subtracting the equations from one another and using (2.57) yields log a(k) =

J X

log

m=1

log a ¯(k) =

J X

m=1

log



k − km ∗ k − km



+



k − k¯m ∗ k − k¯m



−

1 2πi

Z

1 2πi

Z

+∞ −∞ +∞ −∞

15

log (α(ξ)¯ α(ξ)) dξ, ξ−k

Im k > 0

(2.58)

log (α(ξ)¯ α(ξ)) dξ, ξ−k

Im k < 0

(2.59)

 J¯ J which allows one to recover a(k), a ¯(k) from knowledge of {kj , Im kj > 0}j=1 , k¯j , Im k¯j > 0 j=1 and a(ξ)¯ a(ξ) = (1 + ρ(ξ)¯ ρ(ξ)) Writing

−1

.

log a(k) =

+∞ X

γn

(2.60)

n+1 n=0 (2ik)

P xn+1 and using the well-known expansion for log(1 − x) = − +∞ n=0 n+1 , one can easily obtain from (2.58) and (2.60) " # Z J n+1 ∗ n+1 X (2ikm ) − (2ikm ) 1 +∞ n γn = − (2iξ) log(α(ξ)¯ α(ξ))dξ (2.61) n + 1 π −∞ m=1 which is usually referred to as the trace formula. Note that if r = −q ∗ , then from (2.53)–(2.56) it follows that α(ξ)¯ α (ξ) = a(ξ)a∗ (ξ) and consequently (2.58) can be written as     Z +∞ log 1 + |ρ(ξ)|2 J X 1 k − km dξ, Im k > 0. (2.62) − log a(k) = log ∗ k − km 2πi −∞ ξ−k m=1

2.1.2

Inverse Scattering

The inverse problem consists of constructing a map from the scattering data, i.e. (i) the reflection  J¯ J coefficients ρ(k) and ρ¯(k) on the real axis (ii) the eigenvalues {kj }j=1 and k¯j j=1 and (iii) the J

J¯

norming constants {cj }j=1 and {¯ cj }j=1 , back to the potentials. First we use these data to recover the Jost functions. Then, we recover the potentials in terms of these Jost functions. ¯ (x, k) exist and are analytic In the previous section, we showed that the functions N (x, k) and N 1 in the regions Im k > 0 and Im k < 0 respectively, if q, r ∈ L (R). Similarly, under the same conditions on the potentials, the functions µ(x, k) and µ ¯(x, k) defined by (2.45) are meromorphic in the regions Im k > 0 and Im k < 0, respectively. Therefore, in the inverse problem we assume that the unknown functions are sectionally meromorphic. With this assumption, the equations (2.43)–(2.44) can be considered to be the jump conditions of a Riemann-Hilbert problem. To recover the sectionally meromorphic functions from the scattering data, we convert the RiemannHilbert problem to a system of linear integral equations with the use of Plemelej formula [33]. Case of no poles We begin by solving the Riemann-Hilbert problem in the case where µ and µ ¯ have no poles. Introducing the 2 × 2 matrices
¯ (x, k), µ m+ (x, k) = (µ(x, k), N (x, k)) , m− (x, k) = N ¯(x, k) (2.63) the “jump” conditions (2.43)–(2.44) can be written as

m+ (x, k) − m− (x, k) = m− (x, k)V(x, k)

(2.64)

!

(2.65)

where V(x, k) =

−ρ(k)¯ ρ(k) 2ikx ρ(k)e

16

−ρ¯(k)e−2ikx 0

and m± (x, k) → I as |k| → ∞ in the proper half-plane. We consider the projectors 1 P (f )(k) = 2πi ±

Z

+∞

−∞

f (ζ) dζ ζ − (k ± i0)

(2.66)

such that if f± (k) is analytic in the upper/lower half k-plane and f± (k) → 0 as |k| → ∞ for Im k ≷ 0, then P ± (f∓ )(k) = 0,

P ± (f± )(k) = ±f± (k).

Applying P − to (2.64) yields m− (x, k) = I+

1 2πi

Z

+∞

−∞

m− (x, ξ)V(x, ξ) dξ ξ − (k − i0)

(2.67)

which allows one, in principle, to find m− (x, k). Note that as |k| → ∞ m− (x, k) = I−

1 2πik

Z

+∞

m− (x, ξ)V(x, ξ)dξ + O(k −2 )

(2.68)

−∞

and taking into account the asymptotics (2.22)–(2.25) and the definitions (2.64)–(2.65), eq. (2.68) gives the reconstruction of the potentials in terms of the scattering data, i.e. Z 1 +∞ ρ(k)e2ikx N (2) (x, k)dk (2.69) r(x) = π −∞ Z 1 +∞ ¯ (1) (x, k)dk. q(x) = ρ¯(k)e−2ikx N (2.70) π −∞ Case of poles Suppose now that the potential is such that a(k) and a ¯(k) have a finite number of simple zeros J in the regions Im k > 0 and Im k < 0, respectively, which we denote as {kj , Im kj > 0}j=1 and  J¯ k¯j , Im k¯j < 0 . We shall also assume that a(ξ) 6= 0, a ¯(ξ) 6= 0 for any ξ ∈ R. As before, we j=1

apply P − to both sides of (2.43) and P + to both sides of (2.44). Taking into account the analytic ¯ µ and µ properties of N , N, ¯ and the asymptotics (2.22)–(2.25) and using (2.49), we obtain ¯ (x, k) = N

  X Z +∞ J 1 ρ(ξ)e2iξx N (x, ξ) Cj e2ikx Nj (x) 1 + dξ + 0 (k − kj ) 2πi −∞ ξ − (k − i0) j=1

  X Z +∞ J¯ ¯ −2ikx ¯ ¯ (x, ξ) Cj e Nj (x) 1 ρ¯(ξ)e−2iξx N 0 N (x, k) = + dξ − 1 2πi −∞ ξ − (k − i0) (k − k¯j ) j=1

(2.71)

(2.72)

¯j (x) = N ¯ (x, k¯j ), and we introduced where Nj (x) = N (x, kj ), N Cj =

cj , a0 (kj )

C¯j =

17

c¯j a ¯0 (k¯j )

(2.73)

with 0 denoting derivative with respect to the spectral parameter k. Note that the equations ¯ (x, k) now depend on the extra terms {Nj (x)}J defining the inverse problem for N (x, k) and N j=1  ¯ ¯l (x) J . In order to close the system, we evaluate eq. (2.71) at k = k¯l for any l = 1, . . . , J, ¯ and N l=1 and (2.72) at k = kj for any j = 1, . . . , J, thus getting ¯l (x) = N

Nj (x) =

  X Z +∞ J ¯ 1 Cj e2ikj x Nj (x) ρ(ξ)e2ikl ξ N (x, ξ) 1 + + dξ 0 2πi −∞ (k¯l − kj ) ξ − k¯l j=1

  X Z +∞ J¯ ¯ ¯ ¯ (x, ξ) ¯m (x) ρ¯(ξ)e−2ikm ξ N 1 Cm e−2ikm x N 0 − dξ. + 1 2πi −∞ ξ − km (kj − k¯m ) m=1

(2.74)

(2.75)

The equations (2.71)–(2.72) and (2.74)–(2.75) constitute a linear algebraic-integral system of equa¯ (x, k). tions which, in principle, solve the inverse problem for the eigenfunctions N (x, k) and N By comparing the asymptotic expansions at large k of the right-hand sides of (2.71) and (2.72) to the expansions (2.23) and (2.24), respectively, we obtain r(x) = −2i

J X

(2)

e2ikj x Cj Nj (x) +

j=1

q(x) = 2i

J¯ X

¯

(1)

¯ (x) + e−2ikj x C¯j N j

j=1

1 π

Z

+∞

1 π

Z

+∞

ρ(ξ)e2iξx N (2) (x, ξ)dξ

(2.76)

¯ (1) (x, ξ)dξ ρ¯(ξ)e−2iξx N

(2.77)

−∞

−∞

which reconstruct the potentials and thus complete the formulation of the inverse problem (as before, the superscript ` denotes the `-th component of the corresponding vector). If the potentials decay rapidly enough at infinity, so that ρ(k) can be analytically continJ ued above all poles {kj , Im kj > 0}j=1 and ρ¯(k) can be analytically continued below all poles  J¯ k¯j , Im k¯j < 0 , then the system of equations (2.71)–(2.72) and (2.74)–(2.75) can be simplified j=1

as follows

  1 + 0   0 N (x, k) = − 1

¯ (x, k) = N

1 2πi 1 2πi

Z

C0

Z

¯0 C

ρ(ξ)e2iξx N (x, ξ) dξ ξ−k ¯ (x, ξ) ρ¯(ξ)e−2iξx N dξ ξ−k

(2.78) (2.79)

where C0 is a contour from −∞ to +∞ that passes above all zeros of a(k) and C¯0 is a contour from −∞ to +∞ that passes below all zeros of a ¯(k). In the same hypothesis, eqs. (2.76)–(2.77) can be written as Z 1 r(x) = ρ(ξ)e2iξx N (2) (x, ξ)dξ (2.80) π C0 Z 1 ¯ (1) (x, ξ)dξ. q(x) = ρ¯(ξ)e−2iξx N (2.81) π C¯0 Gel’fand-Levitan-Marchenko equations We can also provide a reconstruction for the potentials by developing the Gel’fand-Levitan-Marchenko integral equations, instead of using the projection operators (cf. [1]). Indeed, let us represent the

18

eigenfunctions in terms of triangular kernels   Z +∞ 0 K(x, s)e−ik(x−s) ds + N (x, k) = 1 x   Z +∞ 1 ¯ ¯ + N (x, k) = K(x, s)eik(x−s) ds 0 x Applying the operator

1 2π

R +∞ −∞

¯ K(x, y) +

s > x, s > x,

Im k > 0 Im k < 0.

(2.82) (2.83)

dk e−ik(x−y) for y > x to the equation (2.78), we get   Z +∞ 0 K(x, s)F (s + y)ds = 0 F (x + y) + 1 x

(2.84)

where F (x) =

1 2π

Z

ρ(ξ)eiξx dξ =

C0

Analogously, operating on eq. (2.79) with

1 2π

1 2π

Z

+∞

ρ(ξ)eiξx dξ − i

−∞

R +∞ −∞

J X

Cj eikj x .

(2.85)

j=1

dk eik(x−y) for y > x gives

  Z +∞ 1 ¯ ¯ K(x, y) + K(x, s)F¯ (s + y)ds = 0 F (x + y) + 0 x

(2.86)

where 1 F¯ (x) = 2π

Z

C¯0

ρ¯(ξ)e−iξx dξ =

1 2π

+∞

Z

ρ¯(ξ)e−iξx dξ + i

−∞

J¯ X

¯ C¯j e−ikj x .

(2.87)

j=1

Eqs. (2.84) and (2.86) constitute the Gel’fand-Levitan-Marchenko equations. Inserting the representations (2.82)–(2.83) for the eigenfuctions into the eqs. (2.80)–(2.81) we obtain the reconstruction of the potentials in terms of the kernels of GLM equations, i.e. ¯ (2)(x, x) r(x) = −2K

q(x) = −2K (1) (x, x),

(2.88)

¯ (j) for j = 1, 2 denote the j-th component of the vectors K and K ¯ where, as usual, K (j) and K respectively. If the symmetry r = ∓q ∗ holds, then, taking into account (2.53)–(2.56), it is easy to check that F¯ (x) = ∓F ∗ (x)

(2.89)

and consequently ¯ K(x, y) =



K (2) (x, y) ∓K (1) (x, y)

∗

.

(2.90)

In this case the eqs. (2.84)–(2.86) solving the inverse problem reduce to K (1) (x, y) = ±F ∗ (x + y) ∓

Z

+∞

x

ds

Z

+∞

ds0 K (1) (x, s0 )F (s + s0 )F ∗ (y + s)

x

and the potentials are reconstructed by means of the first of (2.88). 19

(2.91)

2.1.3

Time evolution

We now show how the time evolution of solutions of NLS can be obtained. We do this by calculating the evolution of the scattering data. Then, by the method of the previous section, one can reconstruct the evolution of the solution. The operator (2.2) determines the evolution of the Jost functions, which can be written as   A B ∂t v = v (2.92) C −A where B, C → 0 as x → ±∞ (since we have assumed that q, r → 0 as x → ±∞). Then the time dependence must asymptotically satisfy   A∞ 0 ∂t v = v as x → ±∞ (2.93) 0 −A∞ where we introduced A∞ = lim A(x, k) = 2ik 2 .

(2.94)

|x|→∞

The system (2.93) has solutions that are linear combinations of    A t 0 e ∞ − + . , v = v = e−A∞ t 0 However, such solutions are not compatible with the fixed boundary conditions of the Jost functions (2.5)–(2.6). We define time-dependent functions ¯ t) ¯ Φ(x, t) = e−A∞ t φ(x; ¯ t) ¯ Ψ(x, t) = eA∞ t ψ(x;

Φ(x, t) = eA∞ t φ(x, t), Ψ(x, t) = e

−A∞ t

ψ(x, t),

(2.95) (2.96)

to be solutions of the time-differential equation (2.92). Then the evolution equations for φ and φ¯ become     A + A∞ B A − A∞ B ¯ φ¯ (2.97) φ, ∂t φ = ∂t φ = C −A + A∞ C −A − A∞ so that, taking into account the eqs.(2.30)–(2.31) and evaluating (2.97) as x → +∞ one gets ∂t a(k) = 0,

∂t a ¯(k) = 0

∂t b(k) = −2A∞ b(k),

∂t¯b(k) = 2A∞¯b(k)

or explicitly a(k, t) = a(k, 0), b(k, t) = e4ik

2

a ¯(k, t) = a ¯(k, 0) t ¯b(k, t) = e−4ik2 t¯b(k, 0). b(k, 0),

(2.98) (2.99)

From (2.98) it is clear that the eigenvalues (i.e. the zeros of a(k) and a ¯(k)) are constant as the solution evolves. Not only the number of eigenvalues, but also their locations are fixed. Thus, the eigenvalues are time-independent discrete states of the evolution. In fact, this time invariance is the underlying mechanism of the elastic soliton interaction in NLS (1.1) and in integrable soliton equations in general. On the other hand, the evolution of the reflection coefficients is given by 2

ρ(k, t) = ρ(k, 0)e4ik t ,

ρ¯(k, t) = ρ¯(k, 0)e−4ik 20

2

t

(2.100)

and this gives also the evolution of the norming constants 2

2 C¯j (t) = C¯j (0)e−4ik t .

Cj (t) = Cj (0)e4ik t ,

(2.101)

The expressions for the evolution of the scattering data allow one to solve the initial-value problem for NLS. The procedure is the following: (i) the scattering data are calculated from the initial time (e.g. at t = 0) according to the procedure illustrated in Sec. 2.1.1; (ii) the scattering data at later time (say, t = t1 ) is determined by the formulae (2.98)–(2.101); (iii) the solution at t = t1 is constructed from the scattering data using, for instance, the reconstruction formulae (2.71)–(2.77).

2.2

Soliton Solutions

In the case where the scattering data comprise proper eigenvalues, but ρ(ξ) = ρ¯(ξ) = 0 for ξ ∈ R, the algebraic-integral system (2.71)–(2.72) and (2.74)–(2.75) reduces to the linear algebraic system ¯l (x) = N

  X J Cj e2ikj x Nj (x) 1 + 0 (k¯l − kj )

(2.102)

j=1

  X J¯ ¯ ¯ ¯m (x) Cm e−2ikm x N 0 + Nj (x) = . ¯ 1 (kj − km )

(2.103)

m=1

The one soliton solution is obtained for J = J¯ = 1. In the relevant physical case, when symmetry r = −q ∗ holds, taking into account (2.52) and (2.56), we get " #−1 ∗ 2 C1∗ |C1 | e2i(k1 −k1 )x −2ik1∗ =− 1− e 2 k1 − k1∗ (k1 − k1∗ ) #−1 " ∗ 2 |C1 | e2i(k1 −k1 )x (2) N1 (x) = 1 − 2 (k1 − k1∗ ) (1) N1 (x)

(1)

(2)

where we used the notation N1 (x) = (N1 (x), N1 (x))T . Then from (2.77) it follows q(x) = −2iη

C1∗ −2iξx e sech(2ηx − 2δ) |C1 |

(2.104)

where e2δ =

k1 = ξ + iη,

|C1 | 2η

(2.105)

and, taking into account the time dependence of C1 as given by (2.101), one finally gets the well known one soliton solution of scalar NLS equation q(x, t) = 2ηe−2iξx+4i(ξ

2

−η 2 )t−i(ψ0 +π/2)

sech(2ηx − 8ξηt − 2δ0 )

(2.106)

with e2δ0 =

|C1 (0)| , 2η

ψ0 = arg C1 (0).

21

(2.107)

Note that the velocity of this solution is given by 4ξ and its amplitude by 2η. Therefore, unlike the KdV solitons, in the NLS equation the height and the speed of a soliton may each be specified independently. This means that it is also possible to construct a solution having two (or more) peaks with different amplitudes, but traveling at the same speed. In this case, the peaks will oscillate periodically in amplitude and the separation between peaks will not increase in the long-time limit [6]. Suppose we consider the solution of NLS ( r = −q ∗ ) corresponding to the scattering data  kj = ξj + iηj , ηj > 0, Cj }Jj=1 ∪ {ρ(ξ) = 0 for ξ ∈ R

sometimes referred to as reflectionless states. In order to get the pure J-soliton solution, one can in principle solve the linear system (2.102)–(2.103). The problem of a J-soliton collision can be investigated by looking at the asymptotic states as t → ±∞ proceeding in a similar way as in [6] or [12]. If ξj 6= ξl for j 6= l, then for t → ±∞ the potential breaks up into individual solitons of the form (2.106), i.e. q(x, t) ∼

J X

qj± (x, t)

t → ±∞

(2.108)

j=1

with 2

2

±

qj± (x, t) = 2ηj e−2iξj x+4i(ξj −ηj )t−i(ψj

+π/2)

sech(2ηj x − 8ξj ηj t − 2δj± ).

(2.109)

Let us fix the values of the soliton parameters such that ξ1 < ξ2 < · · · < ξJ . Then, as t → −∞ the solitons are distributed along the x-axis in the order corrsponding to ξJ , ξJ−1 , . . . , ξ1 ; the order of the soliton sequence is reversed as t → +∞. In order to determine the result of the interaction between solitons, we trace the passage of the Jost functions through the asymptotic states. We denote the soliton coordinates at the instant of time t by xj (t) (|t| is assumed large enough so that one can talk about individual solitons). If t → −∞ then xJ  xJ−1  · · ·  x1 . The T function φ(x, kj ) has the form φ(x, kj ) ∼ e−ikj x (1, 0) when x  xJ . After passing through the T J-th soliton it will be of the form φ(x, kj ) ∼ aJ (kj )e−ikj x (1, 0) where aJ (k) is the transmission coefficient relative to the J-th soliton. Repeating the argument, we find  −ik x  J Y e j φ(x, kj ) ∼ al (kj ) xj+1  x  xj . 0 l=j+1

Upon passing through the j-th soliton, since the corresponding state is a bound state, we get   J Y 0 al (kj ) xj  x  xj−1 . (2.110) φ(x, kj ) ∼ Sj eikj x l=j+1

On the other hand, starting from x  x1 and proceeding in a similar way we find for the Jost function ψ the following asymptotic behavior   j−1 Y 0 ψ(x, kj ) ∼ al (kj ) xj  x  xj−1 (2.111) eikj x l=1

¯ Then comparing (2.110) where we have used (2.30)–(2.31) and solved for ψ, ψ¯ in terms of φ, φ. and (2.111) and recalling (2.47) and (2.73), we get Sj

J Y

al (kj ) = a0 (kj )Cj

j−1 Y l=1

l=j+1

22

al (kj ).

(2.112)

According to (2.107), we write Sj (t) = 2ηj e4ik

2

t+2δj +iψj

where δj and ψj are real functions of time. Eq. (2.112) yields Cj (t) ∼ 2ηj e4ik

2

t+2δj− +iψj−

j−1 J Y Y 1 am (kj )−1 a (k ) l j a0 (kj ) m=1

t → −∞

l=j+1

where δj− , ψj− denote the asymptotics of the functions δj , ψj as t → −∞. Proceeding in a similar fashion as t → +∞ and taking into account that the order of solitons is reversed, we get Cj (t) ∼ 2ηj e4ik

2

t+2δj+ +iψj+

j−1 J Y 1 Y a (k ) am (kj )−1 l j a0 (kj ) m=j+1

t → +∞

l=1

therefore we conclude that e

2(δj+ −δj− )+i(ψj+ −ψj− )

=

j−1 Y

J Y

al (kj )2

l=1

am (kj )−2

m=j+1

or, explicitly, using the formula (2.62) for the pure 1-soliton transmission coefficient e

2(δj+ −δj− )+i(ψj+ −ψj− )

=

j−1 Y l=1

kj − kl kj − kl∗

 2 Y 2 J ∗ kj − km . kj − km m=j+1

(2.113)

According to (2.107), this last formula provides the phase shift of the j-th soliton on the transition between the asymptotic states t → ±∞ due to the interaction with the others. For instance, in the 2-soliton case we get k1 − k2∗
+ − + − (2.114) δ1 − δ1 = − δ2 − δ2 = log k1 − k2  2  2 k1 − k2∗ k1 − k2 ψ1+ − ψ1− = arg ψ2+ − ψ2− = arg . (2.115) k1 − k2 k1∗ − k2

2.3

Conserved Quantities and Hamiltonian structure

Eqs. (2.98) shows that a(k, t) and a ¯(k, t) are conserved in time. Recalling (2.6) and (2.30) we have a(k) = lim φ(1) (x, k)eikx x→+∞

(2.116)


T where φ = φ(1) , φ(2) satisfies the scattering problem (2.1) with boundary condition (2.5). Eliminating φ(2) from the scattering problem (2.1) and substituting ˆ

φ(1) (x, k) = e−ikx+φ

(2.117)

results in a Riccati equation for γ = φˆx   γ 2ikγ = γ − qr + q . q x 2

23

(2.118)

Because φˆ vanishes as |k| → ∞ (Im k > 0), we may expand γ(x, t) =

+∞ X γn (x, t) n . (2ik) n=1

(2.119)

Substituting this into (2.118) and matching the corresponding power of 2ik yields γ1 = −qr, γ2 = −qrx   n−1 X γn γn+1 = q + γk γn−k . q x k=1

From (2.60) and the fact that φˆ vanishes as x → −∞ it follows that log a(k) =

+∞ X

Γn n (2ik) n=1

(2.120)

where Γn =

Z

+∞

γn (x)dx.

(2.121)

−∞

But log a(k) is time independent (for all k with Im k > 0) so Γn must be time independent as well. Thus, we get an infinite set of (global) constants of the motion, and the first few of them are Z Z Γ1 = − q(x)r(x)dx, Γ2 = − q(x)rx (x)dx (2.122) Z   2 (2.123) −qx (x)rx (x) + (q(x)r(x)) dx Γ3 =

and so on. With the reductions r = ∓q ∗ these constants of the motion can be written as Z Z 2 Γ1 = ± |q(x)| dx, Γ2 = ± q(x)qx∗ (x)dx (2.124) Z   4 (2.125) Γ3 = ±qx (x)qx∗ (x) + |q(x)| dx etc.

It is known that the equations solvable by ITS are completely integrable Hamiltonian systems and IST amounts to a canonical transformation from physical variables to (an infinite set of) actionangle variables. The phase space M0 is an infinite-dimensional real linear space with complex coordinates defined by pairs of functions q(x), r(x). By analogy with finite-dimensional coordinates labelled by a discrete parameter, the variable x may be thought of as a coordinate label. On the algebra of smooth functionals on the phase space M0 one can define (see for instance [32]) a Poisson structure by the following Poisson brackets  Z +∞  δF δG δF δF {F, G} = i dx (2.126) − δq(x) δr(x) δr(x) δq(x) −∞ where the variational derivative is defined according to δF (q, r) = F (q + δq, r + δr) − F (q, r)  Z +∞  δF δF = δq(x) + δr(x) dx. δq(x) δr(x) −∞ 24

Obviously, the bracket (2.126) posseses the basic properties of a Poisson brackets, i.e. it is skewsymmetric {F, G} = − {G, F }

(2.127)

linear, i.e. for any constant a, b {aF + bG, H} = a {F, H} + b {G, H} and it satisfies the Jacobi identity {F, {G, H}} + {H, {F, G}} + {G, {H, F }} = 0.

(2.128)

The bracket (2.126) is the infinite dimensional generalization of the usual Poisson bracket in the phase space R2n with real coordinates pk , qk for k = 1, . . . , n  n  X ∂f ∂g ∂f ∂g . − {f, g} = ∂qk ∂pk ∂pk ∂qk k=1

The coordinates q(x), r(x) themselves may be considered as functionals on M0 . Their variational derivatives are generalized functions δq(x) = δ(x − y), δq(y)

δr(x) = δ(x − y) δq(x)

δr(x) δq(x) = =0 δr(y) δq(y)

(2.129)

(2.130)

where δ(x − y) is the Dirac δ-function. Substituting (2.129) into (2.126) gives {q(x), q(y)} = {r(x), r(y)} = 0 {q(x), r(y)} = iδ(x − y). These formulae also yield δF = −i {F, q(x)} . δr(x)

δF = −i {F, r(x)} , δq(x)

A dynamical system is said to be Hamiltonian if it is possible to identify generalized coordinates q and momenta p, and a Hamiltonian H (p, q, t) such that the equations of motion of the system can be written as ∂q = {q, H} ∂t ∂p = {p, H} ∂t

(2.131) (2.132)

where {, } denotes the Poisson brackets. Eqs. (2.131)–(2.132) are called Hamilton’s equations of motion and the variables (p, q) are called conjugate. The dynamical system (2.3)–(2.4) is Hamiltonian, as may be seen from the identification: coordinates (q) : momenta (p) : Hamiltonian (H) :

q(x, t) r(x, t) Z +∞   2 −i qx rx + (qr) dx −∞

25

(2.133) (2.134) (2.135)

with the canonical brackets {q(x, t), r(y, t)} = iδ(x − y) {q(x, t), q(y, t)} = {r(x, t), r(y, t)} = 0.

(2.136) (2.137)

Note that the Hamiltonian is given by the conserved quantity (2.123). In the case when the initial data satisfy the additional constraints r = ∓q ∗ , we identify coordinates q(x, t) and momenta q ∗ (x, t) and the corresponding Hamiltonian and brackets are given by Z +∞   2 4 H = −i ± |qx | + |q| dx (2.138) −∞

{q(x, t), q ∗ (y, t)} = iδ(x − y) {q(x, t), q(y, t)} = {q ∗ (x, t), q ∗ (y, t)} = 0.

26

(2.139) (2.140)

Chapter 3

Integrable Discrete Nonlinear Schr¨ odinger equation In general, any given discretization of an integrable PDE is most likely to be nonintegrable. Even though the integrable PDE is the compatibility condition of a linear operator pair, there is, in general, no pair of linear equations corresponding to the general discrete system. Furthermore, given a discretization which appears to be integrable - e.g. one for which numerical simulations suggest the elastic interaction of solitons - there is no algorithmic method to construct the associated linear pair from the discrete system. One approach is to start with a discrete scattering problem and a time dependence equation and then compute the discrete (in space) compatibility condition. This compatibility condition may then provide an integrable, discrete (in space) evolution equation. In particular, in order to find the integrable spatial discretization of an integrable PDE, one first discretizes the scattering problem associated with the PDE. Then, by suitably expanding the time-dependence matrix in powers of the scattering parameter, one can derive compatible systems of ODE’s. Exactly this method was used in [10] to construct the integrable discretization of NLS, i.e. IDNLS (1.8) i

d 1 qn = 2 (qn+1 − 2qn + qn−1 ) ± |qn |2 (qn+1 + qn−1 ) . dt h

(3.1)

The natural discretization of (2.1) is   vn+1 − vn −ik qn = vn + O(h2 ) (3.2) rn ik h  T (1) (2) , qn = q(nh) and rn = r(nh). We rewrite this finite difference as where vn = v(nh) = vn , vn   z Qn vn+1 = vn (3.3) Rn z −1 where Qn = hqn ,

Rn = hrn ,

z = e−ikh = 1 − ikh + O(h2 ),

z −1 = eikh = 1 + ikh + O(h2 )

and we drop terms of O(h2 ) and higher. Given the scattering problem (3.3) and the timedependence equation   d iQn Rn−1 − 2i (z − z −1 )2 −i(zQn − z −1 Qn−1 ) vn (3.4) vn = i(z −1 Rn − zRn−1 ) −iRn Qn−1 + 2i (z − z −1 )2 dτ 27

the discrete compatibility condition tions

d dτ vn+1

=

d dτ vm m=n+1




is equivalent to the evolution equa-

d Qn = Qn+1 − 2Qn + Qn−1 − Qn Rn (Qn+1 + Qn−1 ) dτ d −i Rn = Rn+1 − 2Rn + Rn−1 − Qn Rn (Rn+1 + Rn−1 ) . dτ i

(3.5) (3.6)

In order to obtain the IDNLS (3.1) from the system (3.5)–(3.6), we change variables as follows Qn → hqn ,

Rn → hrn ,

τ → h−2 t.

(3.7)

Then the system (3.5)–(3.6) becomes 1 d qn = 2 (qn+1 − 2qn + qn−1 ) − qn rn (qn+1 + qn−1 ) dt h 1 d −i rn = 2 (rn+1 − 2rn + rn−1 ) − qn rn (rn+1 + rn−1 ) dt h i

(3.8) (3.9)

which corresponds to IDNLS (3.1) under the reduction rn = ∓qn∗ . In the limit h → 0, nh → x, the system (3.8)–(3.9) becomes (2.3)–(2.4). Moreover, in this limit the time-dependence equation (3.4) becomes (2.2), i.e. the time dependence equation for NLS. It is noteworthy that the usual form of the time dependence matrix does not have a convergent continuum limit (cf. e.g. [1]), but here we have chosen the gauge so that the continuum limit exists. Also, the discrete scattering problem (3.3) becomes, in the continuum limit, the scattering problem (2.1) associated with NLS. The discrete IST for the system (3.5)–(3.6) - and therefore also for IDNLS (3.1) - is described in detail in the next section.

3.1 3.1.1

The Inverse Scattering Transform for IDNLS Overview

In this section we develop the Inverse Scattering Transform for the IDNLS equation, eq. (3.1) on the infinite lattice. The solution qn of IDNLS, at a fixed time, is the potential of the scattering problem (3.3). We formulate the IST procedure for the somewhat more general system (3.5)–(3.6) and then consider (3.1) as a special case. The treatment of the direct problem given here closely mirrors that in [10] (see also [1]). However, here we allow the potential to have infinite support on the lattice, whereas [10] formulate the direct problem for potentials with finite support. Indeed, we show thatPthe direct problem is +∞ well-posed if Qn and Rn are such that kQn k1 , kRn k1 < ∞, where kQn k1 = n=−∞ |Qn | is the L1 norm. This extension to potentials with infinite support is significant in light of the fact that the soliton solutions are not of finite support, but decay (exponentially) fast as n → ±∞. We formulate the inverse problem as a Riemann-Hilbert boundary value problem, in analogy with the inverse problem for the differential scattering problem associated with NLS given in [3], and from it we construct the Gelfand-Levitan-Marchenko integral equations. The Riemann-Hilbert approach was partially developed in [34] but no actual solutions where calculated explicitly. Here, we show how to obtain solutions concretely, including the soliton solutions, from the RiemannHilbert approach to the inverse problem. Finally, we show how to construct soliton solutions for the scalar system (3.5)–(3.6). In particular, we show that nonsingular soliton solutions can exists even when Rn 6= −Q∗n (i.e. when 28

(3.5)–(3.6) does not reduce to IDNLS). Instead, we identify conditions on the scattering data which are sufficient for the existence of a nonsingular soliton solution. Soliton solutions for which Rn 6= −Q∗n are a generalization of the usual soliton solutions of IDNLS.

3.1.2

Direct scattering problem

Jost functions and summation equations We refer to solutions of the discrete scattering problem (3.3) as eigenfunctions with respect to the parameter z. When the potentials |Qn | , |Rn | → 0 as n → ±∞, the eigenfunctions are asymptotic to the solutions of   z 0 vn . vn+1 = 0 z −1 Therefore it is natural to introduce the eigenfunctions defined by the following boundary conditions     0 1 −n n ¯ as n → −∞ (3.10) , φn (z) ∼ z φn (z) ∼ z 1 0 ψn (z) ∼ z −n

  0 , 1

ψ¯n (z) ∼ z n

  1 0

as n → +∞.

(3.11)

In the following analysis, it is convenient to consider functions with constant boundary conditions. Hence, we define the Jost functions as follows ¯ n (z) = z n φ¯n (z), M ¯n (z) = z −n ψ¯n (z). N

Mn (z) = z −n φn (z), Nn (z) = z n ψn (z),

(3.12) (3.13)

If the scattering problem (3.3) is rewritten as ˜ n vn vn+1 − Avn = Q where A=



z 0 0 z −1



,

˜n = Q



0 Qn Rn 0



,

(3.14)

then the Jost functions are solutions of the difference equations ˜ n Mn (z) Mn+1 (z) − z −1 AMn (z) = z −1 Q ¯ ¯ ¯ n (z) ˜ Mn+1 (z) − zAMn (z) = z Qn M ˜ n Nn (z) Nn+1 (z) − zANn (z) = z Q −1 −1 ¯n+1 (z) − z AN ¯n (z) = z Q ¯n (z) ˜ nN N with the constant boundary conditions   1 , Mn (z) ∼ 0   0 Nn (z) ∼ , 1

  0 1   ¯n (z) ∼ 1 N 0 ¯ n (z) ∼ M

29

(3.15) (3.16) (3.17) (3.18)

as n → −∞

(3.19)

as n → +∞.

(3.20)

In analogy with the continuous case, we use the method of Green’s functions in order to construct a set of summation equations whose solutions satisfy, respectively, the difference equations (3.15)–(3.16) with the appropriate boundary conditions (3.19)–(3.20). The Green’s function corresponding to (3.15) or, equivalently, (3.18), is a solution of the summation equation Gn+1 − z −1 AGn = z −1 δ0,n I

(3.21)

where δ0,n =



n 6= 0 . n=0

0 1

Now, if vn satisfies the summation equation vn = w +

+∞ X

˜ k vk Gn−k Q

(3.22)

k=−∞

where Gn is a solution of (3.21) and w satisfies w − z −1 Aw = 0,

(3.23)

then vn is a solution of the difference equation (3.15) or, equivalently, (3.18). The Green’s function is not unique, and, as we show below, the choice of the Green’s function and the choice of the inhomogeneous term w together determine the Jost function and its analytical properties. To find the Green’s function explicitly, we first note that the equations for the components of (3.21) are uncoupled. Hence, the off-diagonal terms can be set to zero and ! (1) gn 0 Gn = (3.24) (2) 0 gn (j)

where, according to (3.21), gn must satisfy (j)

gn+1 − b(j) gn(j) = z −1 δ0,n

j = 1, 2

(3.25)

with b(1) = 1,

b(2) = z −2 .

(3.26)

(j)

Next, we represent gn and δ0,n as Fourier integrals I 1 pn−1 gˆ(j) (p)dp, gn(j) = 2πi |p|=1

δ0,n =

1 2πi

I

pn−1 dp.

|p|=1

Substituting these integrals into the difference equations (3.25) yields gˆ(j) (p) = z −1

1 p − b(j)

and gn(j) = z −1

1 2πi

I

|p|=1

30

1 pn−1 dp. p − b(j)

(3.27)

Cout

Cin 1 z2

1 z2

1

1

Figure 3.1: The contours C out and C in for the integrals in (3.27) which avoid singularities at p = 0 and p = z −2 (j) The integral in (3.27) depends only whether the pole (1) b (2)is located inside or outside the contour of integration. However, when |z| = 1 we have b = b = 1, i.e. the poles are on the contour |p| = 1. In analogy with the approach used for the continuous scattering problem, we consider contours that are perturbed away from |p| = 1 in order to avoid the singularities. Let C out be a contour enclosing p = 0 and p = b(j) and let C in be a contour enclosing p = 0 but not p = b(j) (see Fig. 3.1). Consequently, we get  I
n−1 1 1 b(j) n≥1 n−1 −1 gn(j),out = z −1 p dp = z (3.28) 2πi C out p − b(j) 0 n≤0

and gn(j),in = z −1

1 2πi

I

C in

1 pn−1 dp = z −1 p − b(j)



0
n−1 − b(j)

n≥1 . n≤0

(3.29)

By substituting one or the other of (3.28) or (3.29) into (3.24), with b(j) given by (3.26), we obtain two Green’s functions satisfying (3.21), i.e.   1 0 ` −1 (3.30) Gn (z) = z θ(n − 1) 0 z −2(n−1) ¯ rn (z) = −z −1 θ(−n) G



1 0

0 z −2(n−1)



(3.31)

where θ(n) is the discrete version of the Heaviside function, i.e. θ(n) =

n X

δ0,k =

k=−∞



1 0

n≥0 . n<0

(3.32)

Taking into account the boundary conditions (3.19)–(3.20) and the relation (3.23) for the inhomogeneous term in (3.22), if Qn , Rn → 0 as n → ±∞ we get the following summation equations 31

¯n (z) for Mn (z) and N Mn (z) =

  +∞ X 1 ˜ k Mk (z) + G`n−k (z)Q 0

(3.33)

k=−∞

¯n (z) = N

  +∞ X 1 ¯k (z). ¯ rn−k (z)Q ˜ kN + G 0

(3.34)

k=−∞

¯ n (z) and Nn (z) A similar approach yields summation equations for M ¯ n (z) = M

  +∞ X 0 ¯ k (z) ¯ `n−k (z)Q ˜ kM + G 1

(3.35)

k=−∞

  +∞ X 0 ˜ k Nk (z) + Grn−k (z)Q Nn (z) = 1

(3.36)

k=−∞

where  z 2(n−1) 0 0 1  2(n−1)  z 0 Grn (z) = −z θ(−n) . 0 1 ¯ `n (z) = z θ(n − 1) G



(3.37) (3.38)

Existence and Analyticity of the Jost functions We now prove the existence and the sectional analyticity of the Jost functions. We will make use of Lemma A1 and Lemma A2 proved in Appendix A. First we consider the Jost function Mn (z). The Neumann series Mn (z) =

∞ X

Cnj (z)

(3.39)

j=0

where Cn0 (z)

  1 = 0

Cnj+1 (z) =

+∞ X

(3.40) ˜ k C j (z), G`n−k (z)Q k

j≥0

(3.41)

k=−∞

is, formally, a solution of the summation equation (3.33). To make this rigorous, we establish a (j) bound on the Cn such that the series representation (3.39) converges absolutely and uniformly in n and absolutely and uniformly in z in the region |z| ≥ 1. The component-wise summation equation for Mn (z) is Mn(1) (z) = 1 + z −1

n−1 X

(2)

Qk Mk (z)

(3.42)

k=−∞

Mn(2) (z) =

n−1 X

(1)

z −2(n−k−1)−1 Rk Mk (z)

k=−∞

32

(3.43)

and eq. (3.41) in component form is Cnj+1,(1) (z) = z −1

n−1 X

j,(2)

Qk Ck

(z),

Cnj+1,(2) (z) = z −1

n−1 X

j,(1)

z −2(n−1−k) Rk Ck

(z). (3.44)

k=−∞

k=−∞ (0),2

2j+1,(1)

2j,(2)

Then, since Cn = 0, it follows that also Cn = 0, Cn = 0 for any j ≥ 0 and we only 2j,(1) 2j+1,(2) need to find bounds for Cn and Cn . We prove by induction that for |z| ≥ 1 P j j+1 P n−1 n−1 |Q | |R | k k k=−∞ k=−∞ 2j+1,(2) (z) ≤ (3.45) Cn (j + 1)! j! j+1 j+1 P P n−1 n−1 |Q | |R | k k k=−∞ k=−∞ 2(j+1),(1) . (3.46) (z) ≤ Cn (j + 1)! (j + 1)! The inductive step is the application of (3.44) twice. One iteration yields j j P P k−1 k−1 n−1 |Q | |R | X ` ` `=−∞ `=−∞ 2j+1,(2) (z) ≤ |Rk | Cn j! j! k=−∞ j j P P k−1 n−1 n−1 |R | |Q | X ` k `=−∞ k=−∞ |Rk | ≤ j! j! k=−∞

then, applying Lemma A.2 completes the induction for (3.45). The next iteration of (3.44) yields for |z| ≥ 1 n−1 X 2j+1,(2) 2j+2,(1) |Qk | Ck (z) ≤ Cn k=−∞

≤

n−1 X

|Qk |

k=−∞

≤

P

n−1 k=−∞

P

k−1 `=−∞

j!

j P j+1 k−1 |Q` | |R | ` `=−∞

j+1 n−1 |Rk | X

(j + 1)!

k=−∞

(j + 1)!

|Qk |

P

k−1 `=−∞

j!

j |Q` |

and we again use Lemma A2 to complete the induction. The bounds (3.45)–(3.46) are absolutely and uniformly (in n) summable if kQk1 , kRk1 < ∞ where k·k1 is the L1 norm. Moreover, in this case, the Neumann series (3.39) converges uniformly for all |z| ≥ 1. Hence, the Jost function Mn (z) exists and is continuous in the region |z| ≥ 1. Note that we originally constructed Mn (z) for |z| = 1 but the summation equation allows us to extend Mn (z) to the region |z| ≥ 1. Because the Neumann series (3.39) converges absolutely, this yields a convergent Laurent series in powers of z −1 for the solution Mn (z). Thus, the bounds (3.45)–(3.46) estabilish that Mn (z) is analytic in the region |z| > 1 when Qn , Rn ∈ L1 . ¯n (z), it is convenient to In order to prove the existence and analyticity of the Jost function N rewite the difference equation (3.18) as ¯n − zA−1 N ¯n+1 = −z Q ¯n+1 , ˜ nN (1 − Qn Rn )N 33

˜ n given in (3.14), and to define the modified Jost function with A and Q ˆn (z) = cn N ¯n (z) N

(3.47)

where cn =

+∞ Y

(1 − Qk Rk ).

(3.48)

k=n

Note that the product cn converges absolutely if kQk1 , kRk1 < ∞. The modified Jost function ˆn (z) must satisfy the difference equation N ˆn − zA−1 N ˆn+1 = −z Q ˆn+1 ˜ nN N

(3.49)

with the boundary condition ˆn (z) → N

  1 0

as

n → +∞.

(3.50)

By using the method of the Green’s function, we obtain the summation equation ˆn = N

  +∞ X 1 ˆk+1 ˆ rn−k Q ˜ kN + G 0

(3.51)

k=−∞

where ˆ rn (z) = −zθ(−n) G



1 0 0 z −2n



for the solution of the difference equation (3.49) with the boundary condition (3.50). Now, in order to prove the existence of the Jost function Nn (z), we first prove the existence of ˆn (z). As before, we construct the formal Neumann series the modified Jost function N ˆn (z) = N

+∞ X

Cˆnj (z)

(3.52)

j=0

where Cˆn0 (z) = Cˆnj+1 (z) =

  1 0 +∞ X

ˆ rn−k Q ˜ k Cˆ j (z), G k+1

j ≥ 0.

(3.53)

k=−∞ 2j+1,(1)

In the same way as before, one can show by induction that Cˆn and that for |z| ≤ 1

2j,(2) = 0, Cˆn = 0 for any j ≥ 0


j P∞ j+1 P∞ ˆ 2j+1,(2) ( k=n |Rk |) k=n+1 |Qk | (z) ≤ Cn (j + 1)! j!
j+1 P∞ P∞ j+1 ( k=n |Qk |) ˆ 2(j+1),(1) k=n+1 |Rk | (z) ≤ . Cn (j + 1)! (j + 1)! 34

(3.54) (3.55)

The bounds (3.54)–(3.55) guarantee that, if kQk1 , kRk1 < ∞, then the modified Jost function ˆn (z) - i.e., the solution of the summation equation (3.51) and the difference equation (3.49) N ˆn (z) is analytic in the region |z| < 1. exists and is continuous in the region |z| ≤ 1. Moreover, N Then, from (3.47) it follows ¯n (z) = c−1 ˆ N n Nn (z)

(3.56)

ˆn (z) is the solution of the summation equation (3.51) and cn is given by (3.48), provided where N cn 6= 0 for all n. Note that, if Rn = −Q∗n , then Rn Qn ≤ 0 and, therefore, cn > 0 for all n. Alternatively, if |Rn Qn | < 1 for all n then, also, cn > 0 for all n. If, as we already assumed, the potentials are such that kQk1 , kRk1 < ∞, then the condition cn 6= 0 ∀n ∈ Z is equivalent to the condition c−∞ =

+∞ Y

(1 − Qk Rk ) 6= 0

(3.57)

k=−∞

Qn because all partial products k=m (1 − Qk Rk ) are well-defined and finite. Furthermore, cn 6= 0 for all n if, and only if, (1 − Qn Rn ) 6= 0 for all n. Moreover, we will show that the quantity c−∞ is time-independent when the potentials evolve according to (3.5)–(3.6) (cf. [10]). We conclude that, ¯n (z) as defined by (3.56) under the assumptions kQk1 , kRk1 < ∞ and (1 − Qn Rn ) 6= 0 ∀n ∈ Z, N satisfies the difference equation (3.18) with the boundary condition (3.20) and the summation ¯n (z) is continuous in the region |z| ≤ 1 and analytic in equation (3.34). Finally, we note that N |z| < 1 since cn is independent of z. Similar calculations show that Nn (z) exists and is continuous in the region |z| ≥ 1 and that ¯ n (z) exists and is continuous in the region |z| ≤ 1 when kQk , kRk < ∞. Moreover, under the M 1 1 ¯ n (z) is analytic in the region |z| < 1. same condition, Nn (z) is analytic in the region |z| > 1 and M This sectional analyticity of the Jost functions is the key property in the formulation of the inverse problem. Asymptotic behavior of the Jost functions in the complex z-plane Because the Jost functions Mn (z) and Nn (z) are analytic in the region |z| > 1, they have a ¯ n (z) and N ¯n (z) convergent Laurent series expansion about the point z = ∞. Similarly, because M are analytic in the region |z| < 1, they have a convergent power series expansion about z = 0. If we write the leading terms of the Laurent expansions of the components of Mn (z) as Mn(1) = Mn(1),0 + z −1 Mn(1),−1 + O(z −2 ) Mn(2) = Mn(2),0 + z −1 Mn(2),−1 + O(z −2 ) then substituting these expansions into the component-wise summation equations (3.42)–(3.43) and matching the powers of z −1 yields !   (1),0 Mn 1 = (3.58) (2),0 0 Mn (1),−1

Mn (2),−1 Mn

!

=

(2),0

Pn−1

Qk M k (1),0 Rn−1 Mn−1

k=−∞

35

!

=



0 Rn−1



(3.59)

(1),−2

Mn (2),−2 Mn

!

(2),−1

Pn−1

k=−∞ Qk Mk (1),−1 Rn−1 Mn−1

=

!

=

 Pn−1

Qk Rk−1 0

k=−∞



(3.60)

and so on. Moreover, one can show by induction that Mn(2),−2j = 0

Mn(1),−2j−1 = 0,

∀j ≥ 0

(3.61)

and that for any j ≥ 1 n−1 X

Mn(2),−2j+1 =

(1),−2(j+k−n)

(3.62)

(2),−2j+1

(3.63)

Rk Mk

k=n−j n−1 X

Mn(1),−2j =

Qk M k

k=−∞

so that the Laurent expansion of Mn (z) as |z| → ∞ is of the form   1 + O(z −2 , even) Mn (z) = z −1 Rn−1 + O(z −3 , odd)

(3.64)

where “even” indicates that the higher-order terms are even powers of z −1 while “odd” indicates that the higher-order terms are odd powers. ¯ n (z) about z = 0, Analogously, one obtains for the coefficients of the power series expansion of M i.e. ¯ (1) (z) = M n

+∞ X

¯ (1),j , zj M n

n−1 X

¯ (2) (z) = M n

j=0

¯ (2) zj M n

(3.65)

k=−∞

the following relations ¯ (1),0 = 0, M n

¯ (2),0 = 1 M n

(3.66)

and for any integer j ≥ 0 ¯ n(1),2j = 0, M

¯ n(2),2(j+1) = 0 M

and ¯ (1),2j−1 = M n

n−1 X

(2),2(j−n+k)

(3.67)

(1),2j−1

(3.68)

¯ Qk M k

k=n−j

¯ (2),2j = M n

n−1 X

¯ Rk M k

.

k=−∞

Therefore ¯ n (z) = M



zQn−1 + O(z 3 , odd) 1 + O(z 2 , even)



z→0

(3.69)

¯n (z) as well as A similar procedure gives the leading terms of the power series expansion of N the structure of the higher-order terms. However, instead of deriving the power series expansion 36

¯n (z) directly, we derive the power series expansion of the modified Jost function N ˆn (z). In the of N same manner as before, one can show that   1 + O(z 2 , even) ˆ Nn (z) = −zRn + O(z 3 , odd) ¯n (z) = c−1 ˆ around z = 0. Recalling that N n Nn (z) where cn , given by (3.48) is independent of z and by assumption cn 6= 0. Therefore, we also have   2 c−1 n + O(z , even) ¯n (z) = N (3.70) 3 −zc−1 n Rn + O(z , odd) around z = 0. Similar calculations show that   −1 −1 −z cn Qn + O(z −3 , odd) Nn (z) = 2 c−1 n + O(z , even)

|z| → ∞.

(3.71)

Scattering Data The two eigenfunctions with fixed boundary conditions as n → −∞ are linearly independent, as are the two eigenfunctions with fixed boundary conditions as n → +∞. We show this by calculating the Wronskian of these solutions. Let W (v, w) = det |v, w| = v (1) w(2) − v (2) w(1) for any two vectors v = (v (1) , v (2) )T and w = (w(1) , w(2) )T . The vector-valued sequences vn and wn are linearly independent if W (vn , wn ) 6= 0 for all n. In particular, if vn and wn are any two solutions of the scattering problem (3.3), their Wronskian satisfies the recursive relation W (vn+1 , wn+1 ) = (1 − Rn Qn ) W (vn , wn ) therefore for any integer s ≥ 0
W φn (z), φ¯n (z) = =

( (

n−1 Y

k=n−s n−1 Y

k=n−s

)


(1 − Rk Qk ) W φn−s (z), φ¯n−s (z) )


¯ n−s (z) (1 − Rk Qk ) W Mn−s (z), M

and in the limit as s → +∞ we get n−1 Y
W φn (z), φ¯n (z) = (1 − Rk Qk ).

(3.72)

k=−∞

Recall that we have assumed that Qn and Rn are such that 1 − Qn Rn 6= 0 for all n and therefore (3.72) proves that φn , φ¯n are linearly independent. Similarly, for any integer s ≥ 0 (n+s−1 ) Y

−1 W ψ¯n (z), ψn (z) = W ψ¯n+s (z), ψn+s (z) (1 − Rk Qk ) k=n

=

(n+s−1 Y

−1

(1 − Rk Qk )

k=n

37

)


¯n+s (z), Nn+s (z) W N

and the limit as s → +∞, due to the boundary conditions (3.20), yields Y
+∞ ¯ W ψn (z), ψn (z) = (1 − Rk Qk )−1 .

(3.73)

k=n

The discrete scattering problem (3.3) is a second-order difference equation. Therefore, there are at most two linearly independent solutions for any fixed value of z. Any other solution can be expressed as a linear combination of the first two. Since both the left eigenfunctions, φn (z) and φ¯n (z), and the right eigenfunctions ψn (z) and ψ¯n (z), are pairs of linearly independent solutions, we can write φn (z) and φ¯n (z) as linear combinations of ψn (z) and ψ¯n (z) or vice-versa. The coefficients of these linear combinations depend on z. Hence, the relations φn (z) = b(z)ψn (z) + a(z)ψ¯n (z) φ¯n (z) = a ¯(z)ψn (z) + ¯b(z)ψ¯n (z)

(3.74) (3.75)

hold for any z such that all four eigenfunctions φn (z), φ¯n (z), ψn (z) and ψ¯n (z) exist. In particular, ¯ (3.74)–(3.75) hold on |z| = 1 and
define the scattering coefficients a(z), ¯a(z), b(z) and b(z). ¯ Calculating W φn (z), φn (z) using (3.74)–(3.75) results in
 
W φn (z), φ¯n (z) = a(z)¯ a(z) − b(z)¯b(z) W ψ¯n (z), ψn (z) and using eqs. (3.72)–(3.73) to evaluate it for n → +∞ yields

a(z)¯ a(z) − b(z)¯b(z) = c−∞

c−∞ = lim cn = n→−∞

+∞ Y

(3.76)

(1 − Rk Qk ).

(3.77)

k=−∞

This is significant difference with respect to the continuous problem, where the characterization equation (2.32) does not depend on the potentials. Note that we can also introduce the “right” scattering data ψn (z) = d(z)φn (z) + c(z)φ¯n (z) ¯ φ¯n (z), ψ¯n (z) = c¯(z)φn (z) + d(z)

(3.78) (3.79)

which, according to (3.74)–(3.75) and (3.76), are related to the “left” data through c(z) =

+∞ Y

(1 − Rk Qk )−1 a(z)

c¯(z) =

k=−∞

d(z) = −

+∞ Y

+∞ Y

(1 − Rk Qk )−1 a ¯(z)

(3.80)

k=−∞ −1 ¯

(1 − Rk Qk )

¯ =− d(z)

b(z)

+∞ Y

(1 − Rk Qk )−1 b(z).

(3.81)

k=−∞

k=−∞

The scattering coefficients can be represented as Wronskian of the Jost functions. Indeed, using (3.74)–(3.75), it follows

¯n (z), Mn (z) b(z) = cn W ψ¯n (z), φn (z) = z 2n cn W N (3.82)

¯b(z) = cn W φ¯n (z), ψn (z) = z −2n cn W M ¯ n (z), Nn (z) (3.83) 38

a(z) = cn W (φn (z), ψn (z)) = cn W (Mn (z), Nn (z))

¯n (z), M ¯ n (z) a ¯(z) = cn W ψ¯n (z), φ¯n (z) = cn W N

(3.84) (3.85)

where cn is given by (3.48). Recalling the analytic properties of the Jost functions, the expressions (3.84)–(3.85) show that a(z) has an analytic extension in the region |z| > 1 while a ¯(z) has an analytic extension in the region |z| < 1. Because the Jost functions are continuous up to |z| = 1, the functions a(z) and a ¯(z) are also continuous up to |z| = 1. Moreover, substituting the z expansions of the Jost functions (3.64),(3.71) and (3.70)–(3.69) into (3.84)–(3.85) immediatly yields a(z) = 1 + O(z −2 , even) 2

a ¯(z) = 1 + O(z , even)

as |z| → ∞ as |z| → 0.

(3.86) (3.87)

The scattering coefficients can be written as explicit sums of the Jost functions. The formulae are derived as follows. First, we obtain the relation   +∞ n o X 1 − a(z) ¯k (z)a(z) ¯ ˜ k Mk (z) − G ¯ rn−k (z)Q ˜ kN + G`n−k (z)Q Mn (z) − Nn (z)a(z) = 0 k=−∞

by substituting the right-hand sides of the summation equations (3.33) and (3.34) for Mn (z) and ¯n (z) respectively. Then, we use the identity N   0 ¯ rn (z) + z −1 1 −2(n−1) G`n (z) = G 0z to get ¯n (z)a(z) = Mn (z) − N  +



1 − a(z) 0 +∞ X

k=−∞



+

+∞ X

k=−∞

 ¯k (z)a(z) ¯ rn−k (z)Q ˜ k Mk (z) − N G

z −1 0 0 z −2(n−k)+1



˜ k Mk (z). Q

¯n (z)a(z) with z −2n Nn (z)b(z) on both sides Now, by the relation (3.74), we can replace Mn (z) − N r −2n r ¯ and use the identity Gn (z) = z Gn (z) to obtain ( ) +∞ X −2n r ˜ z Nn (z) − Gn−k (z)Qk Nk (z) b(z) = k=−∞



1 − a(z) 0



 +∞  −1 X z 0 ˜ k Mk (z). + Q 0 z −2(n−k)+1 k=−∞

Assuming that the summation equation (3.36) for Nn (z) is uniquely solvable, the term in curly T braces is (0, 1) and then we have a(z) = 1 +

+∞ X

(2)

z −1 Qk Mk (z)

(3.88)

k=−∞

b(z) =

+∞ X

(1)

z 2k+1 Rk Mk (z).

k=−∞

39

(3.89)

The same approach works for a ¯(z) and ¯b(z) and the corresponding expressions are: a ¯(z) = 1 +

+∞ X

¯ (1) (z) zRk M k

(3.90)

k=−∞

¯b(z) =

+∞ X

(2)

¯ (z). z −2k−1 Qk M k

(3.91)

k=−∞

¯ n (z) with In the formulation of the inverse problem, we replace the Jost functions Mn (z) and M the functions   Mn (z) 1 + O(z −2 ) µn (z) = = (3.92) z −1 Rn−1 + O(z −3 ) a(z) µ ¯n (z) =

¯ n (z)  zQn−1 + O(z 3 )  M = . 1 + O(z 2 ) a ¯(z)

(3.93)

Observe that µn (z) is meromorphic in the region |z| > 1 with poles corresponding to the zeros of a(z), while µ ¯n (z) is meromorphic in the region |z| < 1 with poles at the zeros of a ¯(z). Also, we define the reflection coefficients ρ(z) =

b(z) , a(z)

ρ¯(z) =

¯b(z) a ¯(z)

(3.94)

so that the conditions (3.74)–(3.75) are equivalent to ¯n (z) = z −2n ρ(z)Nn (z) µn (z) − N ¯n (z). µ ¯n (z) − Nn (z) = z 2n ρ¯(z)N

(3.95) (3.96)

Note that (3.94), as well as the relations (3.95)–(3.96), are defined, in the general case, only for |z| = 1. Proper Eigenvalues and Norming Constants The discrete scattering problem (3.3) can possess discrete eigenvalues (bound states). These occurr whenever a(zj ) = 0 for some zj such that |zj | > 1 or a ¯(¯ z` ) = 0 for z¯` with |¯ z` | < 1. Indeed, for such values of the spectral parameter W (φn (zj ), ψn (zj )) = 0 and W (φ¯n (¯ z` ), ψ¯n (¯ z` )) = 0 and therefore φn (zj ) and ψn (zj ) and φ¯n (¯ z` ) and ψ¯n (¯ z` ) are linearly dependent, i.e. φ¯n (¯ z` ) = ¯b` ψ¯n (¯ z` )

φn (zj ) = bj ψn (zj ),

(3.97)

for some complex constants bj , ¯b` . In terms of the Jost functions, (3.97) can be written as ¯ n (¯ ¯n (¯ M z` ) = ¯b` z¯`2n N z` )

Mn (zj ) = bj zj−2n Nn (zj ),

(3.98)

which hold if, and only if, zj and z¯` are eigenvalues. Note that the boundary conditions (3.10) and (3.11), together with (3.97), imply that φn (zj ), φ¯n (¯ z` ) → 0 for |n| → ∞. We will assume that neither a(z) nor a ¯(z) vanish on the unit circle. Since the eigenvalues correspond to zeros of the sectionally analytic functions a(z) and a ¯(z), there are no accumulation points of eigenvalues in the regions of analyticity, i.e. |z| > 1 and |z| < 1 respectively. Moreover, because a(z) → 1 as |z| → ∞ and a ¯(z) → 1 as z → 0, there is only a finite number of eigenvalues 40

in the regions |z| ≤ A−1 and |z| ≥ A for any A > 1. If there were an infinite number of zeros of either a(z) or of a ¯(z) (i.e. an infinite number of eigenvalues), these zeros would necessarily have an accumulation point on |z| = 1. Recall, however,that a(z) and a ¯(z) are continuous, respectively, in the regions |z| ≥ 1 and |z| ≤ 1. Hence, the accumulation of zeros at a point on |z| = 1 implies that there is a zero of a(z) or of a ¯(z) on |z| = 1, which contradicts our previous assumption. We conclude that, under the generic assumption a(z), a ¯(z) 6= 0 on |z| = 1, there is a finite number of eigenvalues. We further assume that the eigenvalues are simple zeros of a(z) and a ¯(z). If the eigenvalues are not simple zeros, one can study the situation by the coalescence of simple poles, in analogy with the continuous case [6]. J J¯ Let us assume a(z) has J simple zeros {zj }j=1 and a ¯(z) has J¯ simple zeros at the points {¯ zj }j=1 . Then, by eq. (3.92)-(3.93) and (3.97) Res(µn ; zj ) =

Mn (zj ) bj = 0 z −2n Nn (zj ) = zj−2n Cj Nn (zj ) a0 (zj ) a (zj ) j

(3.99)

¯b` ¯ n (¯ M z` ) ¯n (¯ ¯n (¯ = 0 z¯2n N z` ) = z¯`2n C¯` N z` ) 0 a ¯ (¯ z` ) a ¯ (¯ z` ) `

(3.100)

Res(¯ µn ; z¯` ) =

where 0 denotes derivative with respect to the spectral parameter. We refer to Cj and C¯` as the norming constants associated with the eigenvalues zj and z¯` respectively. If the potentials decay rapidly enough as |n| → ∞ such that b(z) and ¯b(z) can be extended off the unit circle in correspondence of the discrete eigenvalues zj and z¯` respectively, then the norming constants are simply ¯b(z` ) b(zj ) Cj = 0 , C¯` = 0 . a (zj ) a ¯ (¯ z` ) Symmetries First of all we note that the expansions of a(z) and a ¯(z) in (3.86)–(3.87) contain only even powers of z −1 and z respectively. Hence, if zj is a zero of a(z) so is −zj and the same holds for a ¯(z). Therefore, the eigenvalues appear in pairs ±zj , ±¯ z` . Let us denote the norming constants associated with the paired poles ±zj as Cj± and, similarly, we label the constants b± j in (3.98) that are associated with + ±zj . Given the expansions (3.64) and (3.71), we conclude that b− j = −bj . On the other hand, 0 a(z) is even, so a (z) is an odd function of z. Thus, Cj− =

b− −bj bj j = = 0 = Cj+ . a0 (−zj ) −a0 (zj ) a (zj )

(3.101)

Since the two norming constants associated with ±zj are equal, we will drop the superscript ± on the norming constant and refer to both norming constants as Cj . Similarly, one can show that C¯ − = C¯ + = C¯` (3.102) `

`

for the norming constants C¯`± associated with ±¯ z` . Moreover, we also have ρ(−z) = −ρ(z),

ρ¯(−z) = −ρ¯(z).

(3.103)

We have therefore estabilished the following result: Symmetry 3.1 All the eigenvalues appear in pairs ±zj (±¯ z` ) and the norming constant associated with −zj (−¯ z` ) is equal to the norming constant associated with +zj (+¯ z` ).  J¯ J J¯ J , The eigenvalues {±zj } , {±¯ zj } and the associated norming constants {Cj } , C¯j j=1

j=1

j=1

together with the reflection coefficients (3.94), constitute the set of the scattering data S(z). 41

j=1

Symmetry Reductions The evolution equations (3.1) are a special case of the compatibility condition (3.8)–(3.9) where Rn = ∓Q∗n . This symmetry in the potentials induces a symmetry betwen the Jost functions analytic in the region |z| < 1 and the Jost functions analytic in the region |z| > 1. In turn, this symmetry of the Jost functions induces a symmetry in the scattering data. Indeed, a direct computation shows that the Green’s functions (3.30)–(3.31) and (3.37)–(3.38) satisfy the identities
−1 ˆ ∓ G`n (1/z ∗ ) ∗ P ˆ∓ = G ¯ `n (z), ˆ ∓ (Grn (1/z ∗ ))∗ P ˆ −1 ¯r P (3.104) P ∓ = Gn (z), where

ˆ ∓= P



0 ∓1 1 0



.

˜ n in (3.14) is such Moreover, under the symmetry reductions Rn = ∓Q∗n the matrix potential Q that ˆ ∓Q ˜ ∗n P ˆ −1 ˜ P ∓ = Qn

(3.105)

ˆ ∓ Mn∗ (1/z ∗ ) and P ˆ ∓ Nn∗ (1/z ∗) satisfy, respectively, the and from (3.104)–(3.105) it follows that P ¯ ¯ same summation equations as Mn (z) and ∓Nn (z), i.e. eqs. (3.35) and (3.34). Therefore, assuming such equations are uniquely solvable, we obtain the symmetries ¯ n (z) = P ˆ ∓ Mn∗ (1/z ∗), M

¯n (z) = ∓P ˆ ∓ Nn∗ (1/z ∗ ) N

(3.106)

or equivalently ˆ ∓ φ∗n (1/z ∗), φ¯n (z) = P

ˆ ∓ ψn∗ (1/z ∗). ψ¯n (z) = ∓P

(3.107)

The symmetry of the scattering coefficients a(z), a ¯(z) and b(z), ¯b(z) follows from the symmetry of the Jost functions and the Wronskian formulae (3.82)-(3.83), i.e. a ¯(z) = a∗ (1/z ∗ ) ¯b(z) = ∓b∗ (1/z ∗ )

(3.108) (3.109)

when Rn = ∓Q∗n . From (3.108)–(3.109) it also follows the symmetry between the reflection coefficients ρ¯(z) = ∓ρ∗ (1/z ∗ ).

(3.110)

Note that (3.109) and (3.110) hold, in general, only for |z| = 1. The symmetry (3.108) implies that the eigenvalues are paired, i.e. for each eigenvalue zj such zj | < 1 and vice-versa. Consequently, it also that |zj | > 1, there is an eigenvalue z¯j = 1/zj∗ with |¯ ¯ follows that J = J. We now compute the symmetry of the norming constants. With the pairing of the eigenvalues, eqs. (3.98) give ¯bj = ∓b∗ . j Moreover, from (3.108) it also follows that
∗ a ¯0 (¯ zj ) = − zj2 a0 (zj ) 42

(3.111)

hence C¯j =

¯bj −b∗j = = ±(zj∗ )−2 Cj∗ a ¯0 (¯ zj ) −(zj2 a0 (zj ))∗

(3.112)

when Rn = ∓Q∗n and the zeros of a(z) and a ¯(z) are simple. Note that when Rn = Q∗n there are no discrete eigenvalues with |zj | = 6 1. To summarize, we have shown that: Symmetry 3.1 All the eigenvalues of the scattering problem (3.3) appear in pairs ±zj with |zj | > 1, i.e. outside the unit circle, or ±¯ z` , with |¯ z` | < 1, and the norming constant associated with −zj (resp. −¯ z` ) is equal to the norming constant associated with +zj (resp. +¯ z` ). Correspondingly, the norming constant associated with the pair of eigenvalues ±zj outside the unit circle will be denoted by Cj and the norming constant associated with the pair of eigenvalues ±¯ z` inside the unit circle will be denoted by C¯` . Symmetry 3.2If the potentials satisfy the symmetry Rn = ∓Q∗n , then the scattering coefficients satisfy the symmetry a ¯(z) = a∗ (1/z ∗ ),

ρ¯(z) = ∓ρ∗ (1/z ∗ ).

(3.113)

It follows that z¯j = 1/zj∗ is an eigenvalue such that |¯ zj | < 1 if, and only if, zj is an eigenvalue such ¯ i.e. the number of eigenvalues inside the unit circle equals the that |zj | > 1. Therefore J = J, number of eigenvalues outside, and, taking also into account Symmetry 3.1, the eigenvalues come in quartets  J (3.114) ±zj , ±1/zj∗ j=1 .

Moreover, according to (3.112), the norming constants associated with these paired eigenvalues satisfy the symmetry C¯j = ±(zj∗ )−2 Cj∗ . (3.115) Trace formula J¯

J

Assume a(z) and a ¯(z) to have the simple zeros {±zj : |zj | > 1}j=1 and {±¯ zj : |¯ zj | < 1}j=1 , respectively, and define J¯ ∗ −2 Y z 2 − (¯ zm ) α ¯ (z) = a ¯(z) 2−z 2 z ¯ m m=1

J ∗ −2 Y z 2 − (zm ) a(z), α(z) = 2 − z2 z m m=1

(3.116)

According to these definitions, the function α(z) is analytic outside the unit circle, whereas it has no zeros, while α(z) ¯ is analytic inside the unit circle, whereas it has no zeros; moreover, due to (3.86), α(z) → 1 as |z| → ∞. Therefore we have I I 1 1 log α(w) log α(w) ¯ log α(z) = − dw, dw = 0 |z| > 1 2πi |w|=1 w − z 2πi |w|=1 w − z I I 1 log α ¯ (w) log α(w) 1 log α ¯ (z) = dw, dw = 0 |z| < 1. 2πi |w|=1 w − z 2πi |w|=1 w − z Subtracting the equations from one another and using (3.116) yields   I J 2 X 1 log (α(w)¯ α(w)) z 2 − zm − log a(z) = log dw, 2 ∗ −2 z − (zm ) 2πi |w|=1 w−z m=1 log a ¯(z) =

J¯ X

m=1

log



2 z 2 − z¯m 2 ∗ z − (¯ zm )−2



+

1 2πi 43

I

|w|=1

log (α(w)¯ α(w)) dw, w−z

|z| > 1

(3.117)

|z| < 1

(3.118)

J¯

J

which allow to recover a(z), a ¯(z) from knowledge of {±zj : |zj | > 1}j=1 , {±¯ zj : |¯ zj | < 1}j=1 and −1

a(w)¯ a(w) = c−∞ (1 − ρ(w)¯ ρ(w)) for |w| = 1. Note that if Rn = −Q∗n , then from (3.108)–(3.111) and (3.76) it follows that α(w)¯ α (w) =  −1 2 c−∞ 1 + |ρ(w)| for |w| = 1 and consequently (3.117) can be written as log a(z) =

J X

m=1

3.1.3

log



2

2 z − zm 2 ∗ z − (zm )−2



+

1 2πi

I

|w|=1

  2 log 1 + |ρ(w)| w−z

dw,

|z| > 1.

(3.119)

Inverse Scattering

The inverse problem consists of reconstructing the potentials in terms of the scattering data S =  J¯ J {ρ(w), ρ(w) ¯ for |w| = 1} ∪ {±zj , Cj }j=1 ∪ ±¯ zj , C¯j j=1 . In the previous section, we showed that ¯n (z) exist and are analytic in the regions |z| > 1 and |z| < 1 respectively, the functions Nn (z) and N if kQk1 , kRk1 < 1. Similarly, under the same conditions on the potentials, the functions µn (z) and µ ¯n (z) defined by (3.92)–(3.93) are meromorphic in the regions |z| > 1 and |z| < 1 respectively. Therefore, in the inverse problem we assume this analytic properties for the unknown functions. With this assumption, the equations (3.95)–(3.96) can be considered to be the jump conditions of a Riemann-Hilbert problem. As in the continuous case, in order to recover the sectionally meromorphic functions from the scattering data, we convert the Riemann-Hilbert problem to a system of linear integral equations with the use of Plemelj formula (cf. [34]). Because the functions µn (z) and µ ¯n (z) can be meromorphic, the jump conditions (3.95)–(3.96) are insufficient to fix the solution of the Riemann-Hilbert problem. We also need information about the residue of the poles. Furthermore, in order to fix the solution, it is also necessary to specify the boundary conditions. Boundary conditions and residues The functions µn (z) and Nn (z) are meromorphic in the region |z| > 1 and have the limits     1 1 Nn (z) → , µ (z) → as |z| → ∞ n c−1 0 n Q (cf. (3.64),(3.71), and (3.86)). Here, as before, cn = ∞ k=n (1 − Qk Rk ). The boundary condition for Nn (z) depends on Qk and Rk for all k ≥ n. However, Qn and Rn are unknowns in the inverse problem. In order to remove this dependence, we introduce the functions    −1 −1  1 0 −z cn Qn Nn0 = Nn = + O(z −2 ) as |z| → ∞ (3.120) 0 cn 1     1 0 1 µ0n = µn = + O(z −2 ) as |z| → ∞ (3.121) 0 cn z −1 cn Rn−1   −1   cn ¯n0 = 1 0 N ¯n = N + O(z 2 ) as z → 0 (3.122) 0 cn −zRn     1 0 zQn−1 µ ¯0n = µ ¯n = + O(z 2 ) as z → 0. (3.123) 0 cn cn These modified functions satisfy modified jump conditions from (3.95)–(3.96) on |z| = 1, i.e. ¯ 0 (z) = z −2n ρ(z)N 0 (z) µ0n (z) − N n n ¯n0 (z). µ ¯0n (z) − Nn0 (z) = z 2n ρ¯(z)N 44

(3.124) (3.125)

Also, the poles of µ0n (z) and µ ¯0n (z) are the same as the poles of µn (z) and µ ¯n (z) respectively and the residues of these poles are determined by the relations ¯ 0 (zj ) Res(¯ µ0n ; z¯j ) = z¯j2n C¯j N n

Res(µ0n ; zj ) = zj−2n Cj Nn0 (zj ),

(3.126)

which follow from (3.99)–(3.100). Case of no poles Let us consider first the case when there are no discrete eigenvalues, i.e. µ0n and µ ¯0n have no poles. Introducing the 2 × 2 matrices
¯n0 (z), µ mn (z) = (µ0n (z), Nn0 (z)) , m ¯ n (z) = N ¯0n (z) (3.127)

with mn (z) analytic outside the unit circle |z| = 1 and m ¯ n (z) analytic inside, the “jump” conditions (3.124)–(3.125) can be written as mn (z) − m ¯ n (z) = m ¯ n (z)Vn (z)

|z| = 1

(3.128)

where Vn (z) =

−z 2n ρ¯(z) 0

−ρ(z)¯ ρ(z) −2n z ρ(z)

!

(3.129)

and mn (z) → I

as |z| → ∞.

(3.130)

Therefore (3.128) can be regarded as a generalized Riemann-Hilbert boundary value problem on |z| = 1 with boundary conditions given by (3.130). We consider the integral operators I 1 f (w) P¯ (f )(z) = lim dw (3.131) ζ→z 2πi |w|=1 w − ζ |ζ|<1

1 P (f )(z) = lim ζ→z 2πi |ζ|>1

I

|w|=1

f (w) dw w−ζ

(3.132)

defined for |z| < 1 and |z| > 1, respectively, for any function f (w) continuous on |w| = 1. Applying P¯ to both sides of equations (3.128) yields I 1 m ¯ n (w)Vn (w) m ¯ n (z) = I − lim dw (3.133) ζ→z 2πi |w|=1 w−ζ |ζ|<1

which allows one, in principle, to find m ¯ n (z). In component form, (3.133) yields   I w−2n ρ(w)¯ µ0n (w) − ρ(w)¯ ρ(w)N¯n0 (w) ¯n0 (z) = 1 − lim 1 N dw 0 ζ→z 2πi |w|=1 w−ζ |ζ|<1   I w2n ρ¯(w)N¯n0 (w) 1 0 dw µ ¯0n (z) = + lim 1 ζ→z 2πi |w|=1 w−ζ |ζ|<1

45

(3.134)

(3.135)

¯ 0 (z) and µ which is a system of linear integral equations, on |z| = 1, for N ¯0n (z) in terms of the n scattering data. The solutions of these integral equations can be analytically continued into the region |z| < 1. Equivalently, applying the outside projector P to both sides of equations (3.124)–(3.125) we get   I 1 w2n ρ¯(w)N¯n0 (w) 0 0 dw + lim Nn (z) = 1 ζ→z 2πi |w|=1 w−ζ |ζ|>1   I 1 w−2n ρ(w)Nn0 (w) 1 0 µn (z) = − lim dw. 0 ζ→z 2πi |w|=1 w−ζ |ζ|>1

To recover Qn and Rn we compute the power series expansions of the Jost functions around z = 0. By comparing the expansions (3.122) and (3.123) with the expansions about z = 0 of the right-hand sides of (3.134) and (3.135), or, equivalently, of (3.133) I z w−2 m ¯ n (w)Vn (w)dw + O(z −2 ) m ¯ n (z) = I− 2πi |w|=1 we obtain I 1 ¯ 0(1) (w)dw w2n ρ¯(w)N n+1 2πi |w|=1 I   1 −2 w−2(n+1) ρ(w)¯ µ0(2) ρ(w)¯ ρ(w)N¯n0(2) (w) dw Rn = n (w) − w 2πi |w|=1

Qn =

¯n0 (z) and µ where N ¯0n (z) are determined by the linear system (3.134)–(3.135) and the superscript (j) for j = 1, 2 denotes the j-th component of the corresponding vector. Hence, the formulation of the inverse problem is complete. Note that we recover the potentials from the Jost functions ¯n0 (z), because there are no terms in which are defined in the region |z| ≤ 1, namely µ ¯0n (z) and N 0 0 the z expansions of µn (z) and Nn (z) which are simple functions of Qn and Rn . Case of poles The method of solution requires an extra step if µ0n (z) and µ ¯0n (z) have poles. As before, we apply ¯ P to both sides of (3.124) and P to both sides of (3.125) to obtain ¯n0 (z) = N

    J X 1 1 1 −2n 0 0 + Cj zj N (zj ) + N (−zj ) 0 z − zj n z + zj n j=1 I 1 w−2n ρ(w)Nn0 (w) − lim dw ζ→z 2πi |w|=1 w−ζ

(3.136)

|ζ|<1

Nn0 (z) =

    J¯ X 1 ¯0 1 ¯0 0 Nn (¯ zj ) + Nn (−¯ zj ) + C¯j z¯j2n 1 z − z¯j z + z¯j j=1 I 1 w2n ρ¯(w)N¯n0 (w) + lim dw ζ→z 2πi |w|=1 w−ζ |ζ|>1

46

(3.137)

where Nn0 (zj ) is Nn0 (z) evaluated at the eigenvalue zj , Nn0 (−zj ) is Nn0 (z) evaluated at the eigenvalue ¯n0 (¯ ¯n0 (−¯ −zj and similarly for N zj ) and N zj ). Here we have included the fact that the eigenvalues arise in pairs ±zj in |z| > 1 and ±¯ z` in |z| < 1 and that the corresponding norming constants satisfy (3.101)–(3.102). The equations (3.136)–(3.137) constitute a system of linear integral equations on  0 J¯ J ¯ (¯ ¯ 0 (−¯ |z| = 1. This system depends on the vectors {N 0 (zj ), N 0 (−zj )} and N zj ), N zj ) . n

n

j=1

n

n

j=1

We obtain expressions for these vectors by evaluating (3.136) at the points ±¯ zj and (3.137) at the points ±zj . This results in a linear algebraic-integral system composed of (3.136)–(3.137) and ¯n0 (¯ N zj ) =

  X   J 1 1 1 −2n 0 0 + Ck zk N (zk ) + N (−zk ) 0 z¯j − zk n z¯j + zk n k=1 I w−2n ρ(w)Nn0 (w) 1 − dw 2πi |w|=1 w − z¯j

¯ 0 (−¯ N zj ) = n

Nn0 (zj ) =

Nn0 (−zj )

    X J 1 1 1 − Ck zk−2n Nn0 (zk ) + Nn0 (−zk ) 0 z¯j + zk z¯j − zk k=1 I 1 w−2n ρ(w)Nn0 (w) − dw 2πi |w|=1 w + z¯j    X  J¯ 1 1 0 ¯n0 (¯ ¯n0 (−¯ C¯k z¯k2n + N zk ) + N zk ) 1 zj − z¯k zj + z¯k k=1 I 2n w ρ¯(w)N¯n0 (w) 1 dw + 2πi |w|=1 w − zj

   X  J¯ 1 1 0 2n 0 0 ¯ ¯ ¯ − Ck z¯k = N (¯ zk ) + N (−¯ zk ) 1 zj + z¯k n zj − z¯k n k=1 I ¯n0 (w) w2n ρ¯(w)N 1 dw + 2πi |w|=1 w + zj

(3.138)

(3.139)

(3.140)

(3.141)

J¯

where (3.138)–(3.139) hold for each eigenvalue {¯ zj }j=1 and (3.140)–(3.141) hold for each eigenvalue J

{zj }j=1 . As in the case where there are no discrete eigenvalues, we can recover Rn from the power series ¯ 0 (z). There is, however, no easy way to obtain the potential from N 0 (z). Instead, expansion of N n n ¯ we apply P to both sides of (3.125) in order to obtain the representation µ ¯0n (z)

   X  J¯ 1 ¯0 1 ¯0 0 2n ¯ + Cj z¯j = N (¯ zj ) + N (−¯ zj ) 1 z − z¯j n z + z¯j n j=1 I ¯ 0 (w) w2n ρ¯(w)N 1 n + dw. 2πi |w|=1 w−z

(3.142)

Now, by comparing the power series expansions of the right-hand sides of (3.136) and (3.142) to

47

the expansions (3.122) and (3.123), respectively, we obtain J X

1 2πi

I

1 2(n−1) ¯ 0(1) = −2 C¯j z¯j Nn (¯ zj ) + 2πi j=1

I

Rn = 2

−2(n+1)

Cj zj

Nn0(2) (zj ) +

j=1

Qn−1

J¯ X

|w|=1

w−2(n+1) ρ(w)Nn0(2) (w)dw

(3.143)

w2(n−1) ρ¯(w)N¯n0(1) (w)dw.

(3.144)

|w|=1

where we have used that for the solutions of the system (3.136)–(3.141) the following relations hold Nn0(1) (−z) = −N 0(1) (z), ¯ 0(1) (−z) = N ¯ 0(1) (z), N

N 0(2) (−z) = N 0(2) (z) ¯ 0(2) (−z) = −N ¯ 0(2) (z). N

n

Note that from (3.123), (3.142) it also follows cn = 1 − 2

J¯ X

1 ¯ 0(2) (¯ C¯j z¯j2n−1 N zj ) + n 2πi j=1

I

|w|=1

¯ 0(2) (w)dw. w2n−1 ρ¯(w)N n

(3.145)

When Rn = Q∗n there are no discrete eigenvalues off the unit circle; using the symmetry relation (3.113) between the reflection coefficients ρ and ρ¯ (cf. Symmetry 3.2), eq.(3.144) can be written as I 1 0(1) w2n ρ∗ (w)N¯n+1 (w)dw. Qn = 2πi |w|=1 Analogously, if the potentials satisfy the symmetry condition Rn = −Q∗n , then the scattering data satisfy relations (3.113)–(3.115) and (3.144) becomes Qn = −2

J¯ X

¯ 0(1) (1/zj∗) − 1 (zj∗ )−2(n+1) Cj∗ N n+1 2πi j=1

I

|w|=1

0(1) w2n ρ∗ (w)N¯n+1 (w)dw.

Reflectionless Potentials In the case where the scattering data comprises proper eigenvalues, but ρ(z) = ρ¯(z) = 0 on |z| = 1, the algebraic-integral system (3.138)–(3.141) reduces to the linear algebraic system ¯n0 (¯ N zj ) =

   X  J 1 1 1 + Ck zk−2n Nn0 (zk ) + Nn0 (−zk ) 0 z¯j − zk z¯j + zk k=1

¯n0 (−¯ N zj ) =

   X  J 1 1 1 −2n 0 0 N (zk ) + N (−zk ) − Ck zk 0 z¯j + zk n z¯j − zk n k=1

Nn0 (zj ) =

   X  J¯ 1 1 0 ¯n0 (¯ ¯n0 (−¯ + C¯k z¯k2n N zk ) + N zk ) 1 zj − z¯k zj + z¯k k=1

Nn0 (−zj )

  X   J¯ 1 1 0 2n 0 0 ¯ ¯ ¯ − Ck z¯k = N (¯ zk ) + N (−¯ zk ) . 1 zj + z¯k n zj − z¯k n k=1

48

Moreover, the potentials are given by Qn−1 = −2

J¯ X

2(n−1) ¯ 0(1) C¯j z¯j Nn (¯ zj )

j=1

Rn = 2

J X

−2(n+1)

Cj zj

Nn0(2) (zj ).

j=1

If there is one quartet of eigenvalues {±z1 , ±¯ z1 } with |z1 | > 1 and |¯ z1 | < 1, then 2(n+1)

Qn = −

2D1 z¯1

2(n+2)

1 + 4C1 D1 (z12 − z¯12 )−2 z1−2n z¯1

(3.146)

−2(n+1)

2C1 z1

Rn =

2(n+2)

1 + 4C1 D1 (z12 − z¯12 )−2 z1−2n z¯1

(3.147)

where we introduced the modified norming constant D1 = z¯1−2 C¯1 .

(3.148)

c−∞ = lim cn = z12 z¯1−2 .

(3.149)

Note that from (3.145) it also follows n→−∞

In order to obtain a nonsingular potential, we impose the following symmetry: Symmetry 3.3 The scattering data are such that (i) the eigenvalues satisfy the relation z¯1 = 1 , (ii) the product of the norming constants C1 and D1 is real and moreover (iii) C1 D1 > 0. ∗ z 1

Then, with the substitution z1 = e(α+iβ) , the expressions (3.146)–(3.147) can be written in the form Qn = − Rn =

D1 (C1 D1 ) C1

1/2

(C1 D1 )

sinh(2α) e2i(n+1)β sech (2α(n + 1) − δ)

(3.150)

sinh(2α) e−2i(n+1)β sech (2α(n + 1) − δ)

(3.151)

1/2

where δ = log (C1 D1 )1/2 − log sinh(2α). Recall that if Rn = −Q∗n , then Symmetry 3.2 holds. Therefore, in particular, if Rn = −Q∗n and the associated scattering data consist of a single quartet of eigenvalues (and their respective norming constants), then, as a consequence, Symmetry 3.3 holds. However, the converse is not necessarily true. The Symmetry 3.3 in the scattering data is insufficient to ensure that (3.146)– (3.147) satisfy the symmetry Rn = −Q∗n . In order to obtain the symmetry Rn = −Q∗n in (3.146)– (3.147) we must impose the condition |C1 | = |D1 | in addition to Symmetry 3.3. Equivalently, in order to obtain the symmetry Rn = −Q∗n we must have (i) z¯1 = z1∗ and (ii) C¯1 = z¯12 C1∗ . 1 Typically, only solutions with the symmetry Rn = −Q∗n are considered. However, we emphasize that the sech profile of the potentials (3.150)–(3.151) results from the Symmetry 3.3 in the scattering data and we do not need to require that Rn = −Q∗n . In the scalar evolution equation i.e. IDNLS - Symmetry 3.3 is only slightly more general that Symmetry 3.2. Neverthless, we have shown that the sech envelope potentials exist in a more general setting (i.e. when Rn 6= −Q∗n ). 49

We recall that, if the potentials are not constrained to satisfy the symmetry Rn = −Q∗n , then, in order to ensure the IST procedure is well-defined, we must separately impose the condition (1 − Rn Qn ) 6= 0 for all n. However, we point out that this second condition is time-invariant if kRk1 , kQk1 < ∞. Gel’fand-Levitan-Marchenko equations Like in the continuous case, we can also provide a reconstruction for the potentials by means of Gel’fand-Levitan-Marchenko integral equations. Indeed, let us represent the eigenfunctions ψn and ψ¯n in terms of triangular kernels ψn (z) =

+∞ X

z −j K(n, j)

|z| > 1

(3.152)

j=n

ψ¯n (z) =

+∞ X

¯ z j K(n, j)

|z| < 1

(3.153)

j=n


T
¯ ¯ (1) (n, j), K ¯ (2) (n, j) T and write eqs. (3.74)– where K(n, j) = K (1) (n, j), K (2) (n, j) , K(n, j) = K (3.75) in the form φn (z)a−1 (z) − ψ¯n (z) = ψn (z)ρ(z) a−1 (z) − ψn (z) = ψ¯n (z)¯ ρ(z) φ¯n (z)¯

(3.154) (3.155)

with ρ and ρ¯ given by (3.94). H 1 dz z −m−1 for m ≥ n to the equation (3.154) and taking into Applying the operator 2πi |z|=1 account the asymptotics (3.64), (3.69) and (3.86)–(3.87), as well as the triangular representations (3.152)–(3.153), we obtain ¯ K(n, m) +

+∞ X

  1 δ 0 m,n

K(n, j)F (m + j) =

j=n

m≥n

(3.156)

where F (n) =

J X

zj−n−1 Cj

j=1

Analogously, operating on eq. (3.155) with K(n, m) +

+∞ X

1 2πi

1 + 2πi H

C

I

z −n−1 ρ(z)dz.

(3.157)

C

dz z −m−1 for m ≥ n yields

¯ K(n, j)F¯ (m + j) =

j=n

  0 δ 1 m,n

m≥n

(3.158)

where F¯ (n) = −

J¯ X j=1

z¯jn−1 C¯j

1 + 2πi

I

z n−1 ρ¯(z)dz.

(3.159)

C

Eqs. (3.156) and (3.158) constitute the Gel’fand-Levitan-Marchenko equations. Note that the sum into (3.157) (resp.(3.159)) are performed over all the discrete eigenvalues which are inside

50

(resp. outside) the unit circle. Since these eigenvalues are paired and the corresponding norming constants satisfy (3.101)–(3.102), the GLM equations can be simplified as follows ¯ K(n, m) +

+∞ X

K(n, j)FR (m + j) =

  1 δ 0 m,n

m≥n

(3.160)

¯ K(n, j)F¯R (m + j) =

  0 δ 1 m,n

m≥n

(3.161)

n = odd n = even

(3.162)

j=n j+m=odd

K(n, m) +

+∞ X

j=n j+m=odd

where  PJ R 1 2 j=1 zj−n−1 Cj + πi z −n−1 ρ(z)dz CR FR (n) = 0 ( R PJ¯ 1 n−1 ρ¯(z)dz −2 j=1 z¯jn−1 C¯j + πi ¯ CR z FR (n) = 0

n = odd n = even

(3.163)

and CR denotes the right half of the unit circle. Comparing the representations (3.152)–(3.153) for the eigenfuctions with the asymptotics (3.70) and (3.71) we obtain, recalling (3.13), the reconstruction of the potentials in terms of the kernels of GLM equations, i.e. ¯ (2) (n, n) = 0, K (1) (n, n) = K Qn = −

K

¯ (1) (n, n) = K (2) (n, n) = c−1 K n (2) ¯ K (n, n + 1) Rn = − ¯ (1) K (n, n)

(1)

(n, n + 1) , K (2) (n, n)

(3.164) (3.165)

¯ (j) for j = 1, 2 are the j-th component of the vectors K and K ¯ where, as usual K (j) and K respectively. It is more convenient to write the equations (3.160)–(3.161) as forced summation equations, which is accomplished if, for m > n we introduce κ (n, m) and κ ¯(n, m) such that     1 0 (3.166) κ ¯(n, n) = κ(n, n) = 0 1 and for m > n K(n, m) =

+∞ Y

(1 − Rj Qj ) κ(n, m)

(3.167)

(1 − Rj Qj ) κ ¯ (n, m).

(3.168)

j=n

¯ K(n, m) =

+∞ Y

j=n

Then equations (3.160)–(3.161) become κ ¯ (n, m) +

  0 FR (m + n) + 1

  1 ¯ κ(n, m) + FR (m + n) + 0

+∞ X

κ(n, j)FR (m + j) = 0

m>n

(3.169)

κ ¯ (n, j)F¯R (m + j) = 0

m>n

(3.170)

j=n+1 j+m=odd +∞ X

j=n+1 j+m=odd

51

and the potentials are obtained from Qn = −κ(1) (n, n + 1),

Rn = −¯ κ(2) (n, n + 1).

(3.171)

If the symmetry Rn = ∓Q∗n holds, then, from (3.107)–(3.111) and (3.112) it follows F¯ (m) = ∓F ∗ (m)  (2) ∗ K (n, m) ¯ K(n, m) = . ∓K (1) (n, m)

(3.172) (3.173)

In this case the eqs. (3.156),(3.158) solving the inverse problem reduce to K

(2)

(n, m) ∓

+∞ X +∞ X

j 0 =n

K (2) (n, j 00 )F (j 0 + j 00 )F ∗ (j 0 + m) = δn,m

(3.174)

j 00 =n

K (1) (n, m) =

+∞ X

K (2) ∗ (n, j 0 )F ∗ (j 0 + m)

(3.175)

j 0 =n

and the potentials are recontructed by means of the first of (3.165).

3.1.4

Time evolution

The operator (3.4) determines the evolution of the Jost functions. From this we deduce the time evolution of the scattering data. Since we have assumed that Qn , Rn → 0 as n → ±∞, then the time dependence (3.4) is asymptotically of the form   −iω 0 vn as n → ±∞ ∂τ vn = 0 iω where ω=


2 1 z − z −1 . 2


T This system has solutions that are linear combinations of the solutions vn+ = e−iωτ , 0 and
T vn− = 0, eiωτ . However, such solutions are not compatible with the fixed boundary conditions of the Jost functions (3.20) and therefore we define the time-dependent functions ¯ n (z, τ ) = eiωτ M ¯ n (z, τ ) M ¯n (z, τ ) N¯n (z, τ ) = e−iωτ N

Mn (z, τ ) = e−iωτ Mn (z, τ ), Nn (z, τ ) = eiωτ Nn (z, τ ),

to be solutions of the time-dependence equation (3.4). These τ -dependent functions satisfy the relations Mn (z, τ ) = z −2n e−2iωτ b(z, τ )Nn (z, τ ) + a(z, τ )N¯n (z, τ ) ¯ n (z, τ ) = z 2n e2iωτ ¯b(z, τ )N ¯n (z, τ ) + a M ¯(z, τ )Nn (z, τ ) which are obtained from the eqs. (3.74)–(3.75).

52

(3.176) (3.177)

To find the expressions for the evolution of the scattering coefficients, we first differentiate (3.176)–(3.177) with respect to τ to obtain ∂τ Mn (z, τ ) = z −2n e−2iωτ {b(z, τ )∂τ Nn (z, τ ) + [bτ (z, τ ) − 2iωb(z, τ )] Nn (z, τ )} ¯n (z, τ ) +aτ (z, τ )N¯n (z, τ ) + a(z, τ )∂τ N

(3.178)

   ¯ n (z, τ ) = z 2n e2iωτ ¯b(z, τ )∂τ N ¯n (z, τ ) + ¯bτ (z, τ ) + 2iω¯b(z, τ ) N ¯n (z, τ ) ∂τ M +¯ aτ (z, τ )Nn (z, τ ) + a ¯(z, τ )∂τ Nn (z, τ ).

(3.179)

¯ n (z, τ ), Nn (z, τ ) and N¯n (z, τ ) satisfy (3.4), On the other hand, because the functions Mn (z, τ ), M we have ¯n (z, τ ) ∂τ Mn (z, τ ) = z −2n e−2iωτ b(z, τ )∂τ Nn (z, τ ) + a(z, τ )∂τ N 2n 2iωτ ¯b(z, τ )∂τ N ¯ n (z, τ ) = z e ¯n (z, τ ) + a ∂τ M ¯(z, τ )∂τ Nn (z, τ ).

(3.180) (3.181)

Comparing (3.178)–(3.179) with, respectively, (3.180)–(3.181) and examining the asymptotics of these expressions as n → +∞, one gets bτ (z, τ ) = 2iωb(z, τ ) a ¯τ (z, τ ) = 0

aτ (z, τ ) = 0 ¯bτ (z, τ ) = −2iω¯b(z, τ )

and therefore b(z, τ ) = b(z, 0)e2iωτ

a(z, τ ) = a(z, 0) ¯b(z, τ ) = ¯b(z, 0)e−2iωτ .

a ¯(z, τ ) = a ¯(z, 0)

(3.182)

The evolution of the reflection coefficients is thus given by ρ(z, τ ) = ρ(z, 0)e2iωτ

(3.183)

ρ¯(z, τ ) = ρ¯(z, 0)e−2iωτ .

(3.184)

From (3.182) it is clear that the eigenvalues (i.e. the zeros of a(z) and a ¯(z)) are constant as the solution evolves. Not only the number of eigenvalues, but also their locations are fixed. Thus, the eigenvalues are time-independent discrete states of the evolution. The norming constants, however, are not fixed. Their evolution is obtained analogously and is given by C¯j (τ ) = C¯j (0)e−2i¯ωj τ

Cj (τ ) = Cj (0)e2iωj τ ,

(3.185)

where ωj =


2 1 zj − zj−1 , 2

ω ¯j =


2 1 z¯j − z¯j−1 . 2

The expressions for the evolution of the scattering data allow one to solve the initial-value problem for IDNLS (3.5)–(3.6). The procedure is the following: (i) the scattering data are calculated from the initial time (e.g. at τ = 0); (ii) the scattering data at later time (say, τ = τ1 ) are determined by the formulae (3.183)–(3.184) and (3.185); (iii) the solution at τ = τ1 is constructed from the scattering data.

53

3.2

Soliton solutions

The soliton solutions of (3.8)–(3.9) are the reflectionless potentials (i.e. ρ(z) = ρ¯(z) = 0 on |z| = 1) where the eigenvalues appear in sets of four. The scattering data of a J-soliton solution are composed of: (i) the 4J eigenvalues ±zj and ±¯ zj = ± z1∗ where |zj | > 1 and j = 1, . . . , J; (ii) j the associated 2J norming constants Cj (τ ), Dj (τ ) where Dj (τ ) = z¯j−2 C¯j (τ ) and Cj (0)Dj (0) ∈ R, Cj (0)Dj (0) > 0. Note that the evolution of the norming constants given by (3.185) assures that Cj (τ )Dj (τ ) ∈ R, Cj (τ )Dj (τ ) > 0 also for τ 6= 0. The one-soliton solution is obtained from (3.150)–(3.151) taking into account the explicit time dependence as given by (3.185), i.e. D1 (0)

ei(2β(n+1)−2wτ ) sinh(2α)sech (2α(n + 1) − 2vτ − δ0 ) (C1 (0)D1 (0))1/2 C1 (0) Rn (τ ) = e−i(2β(n+1)−2wτ ) sinh(2α)sech (2α(n + 1) − 2vτ − δ0 ) 1/2 (C1 (0)D1 (0))

Qn (τ ) = −

(3.186) (3.187)

where v = − sinh(2α) sin(2β), 1/2

δ0 = log (C1 (0)D1 (0))

w = cosh(2α) cos(2β) − 1 − log sinh(2α).

(3.188) (3.189)

Each of the (3.186),(3.187) represents a localized traveling-wave with a single peak that is modulated by a complex carrier phase. This is the one-soliton solution obtained in [10]. Recall that, in order to derive IDNLS (3.5)–(3.6) and the associated pair of linear operators (3.3), (3.4) from NLS (1.1) and the associated pair of linear operators (2.1)–(2.2), we let Qn = hqn = hq(nh),

τ = h−2 t,

Rn = hrn = hr(nh),

z1 = e−ik1 h .

By substituting these expressions into (3.186)–(3.187) we obtain +

˜ ) qn (t) = icAe−2i(ξhn−wt−ψ sech (2ηh(n + 1) − v˜t − δ0 ) −

˜ ) rn (t) = ic−1 Ae2i(ξhn−wt−ψ sech (2ηh(n + 1) − v˜t − δ0 )

(3.190) (3.191)

where A=

sinh(2ηh) , h

w ˜=

1 − cosh(2ηh) cos(2ξh) , h2

v˜ = 2

sinh(2ηh) sin(2ξh) h2

and log |z1 | β arg z1 α = , ξ=− =− , h h h h |D1 (0)|1/2 c= , ψ ± = − arg C1 (0) + 2ξh ± π. 1/2 |C1 (0)| η=

The expressions (3.190)–(3.191) give the one-soliton solution of the version of IDNLS that explicilty contains h, i.e. the system (3.8)–(3.9). In the limit h → 0, nh → x (3.190)–(3.191) become q(x, t) = 2iηce−i(2ξx−4(ξ

2

−η 2 )t−ϕ+ )

sech (2ηx − 8ξηt − θ)

−1 i(2ξx−4(ξ 2 −η 2 )t−ϕ− )

r(x, t) = 2iηc

e

54

sech (2ηx − 8ξηt − θ)

(3.192) (3.193)

θ = lim log h→0

(C1 (0)D1 (0))1/2 sinh(2ηh)

(3.194)

which is the one-soliton solution of (2.5)–(2.6). A similar result holds for the multisoliton solutions. We point out some differences between the discrete soliton and the soliton solution of the PDE. Both (3.190)–(3.191) and (3.192)–(3.193) are composed of traveling-wave with a sech profile which is modulated by a complex carrier wave. Moreover, in both the PDE and the lattice soliton, the velocity of the traveling wave depends on the spatial frequency of the carrier. However, this has different consequences for the lattice and the PDE: 1. on the lattice there is a maximum spatial frequency (i.e. |ξ| ≤ 2π h ) which leads to a maximum sinh(2ηh) speed for the soliton on the lattice (i.e. |v| ≤ ), while there is no such upper bound h2 on soliton velocity in the PDE; 2. in the PDE soliton, the velocity depends linearly on the spatial frequency. In contrast, on the lattice, the velocity goes like sin(2ξh). Hence, the speed increases with |ξ| for small |ξ|, but then decreases to zero as |ξ| further increases. As a consequence, unlike the PDE, it is possible, on the lattice, to have two solitons with the same velocity even though they have different spatial carrier frequencies. Finally, we remark that in the literature the solitons of NLS and IDNLS are generally considered only when r = −q ∗ and rn = −qn∗ . In this case c = 1 and (3.192)–(3.193) give the NLS soliton (2.106). Here, however, we have shown that one can consider the soliton in slightly more general conditions: even though rn 6= −qn∗ , the potentials are still such that (i) the eigenvalues appear in quartets ±zj , ± z1∗ and (ii) the products of the associated norming constants is real and positive j

(see Symmetry 3.3). The characteristic localized traveling-wave solution is a result of a symmetry in the eigenvalues of the solution. As these eigenvalues are time-independent, this symmetry (or lack thereof) is a conserved characteristic of solutions in the non-reduced system (3.190)–(3.191). The problem of a multisoliton collision can be investigated by looking at the asymptotic states as τ → ±∞ proceeding in a similar way as in the continuous case. Consider a pure J-soliton solution for Rn = −Q∗n and assume, without loss of generality, v1 < v2 < · · · < vJ . Then for τ → ±∞ the potential breaks up into individual solitons of the form (3.186) and as τ → −∞ the discrete solitons are distributed along the n-axis in the order corresponding to nJ , nJ−1 , . . . , n1 ; the order of the soliton sequence is reversed as τ → +∞. In order to determine the result of the interaction between solitons, we trace the passage of the Jost functions through the asymptotic states. We denote the soliton coordinates at the instant of time τ by nj (τ ) (|τ | is assumed large enough so that one can talk about individual solitons). If τ → −∞ then nJ  nJ−1  · · ·  n1 . T The function φn (zj ) has the form φn (zj ) ∼ zjn (1 , 0) when n  nJ . After passing through the J-th soliton, it will be of the form φn (zj ) ∼ aJ (zj )zjn (1, 0)T where aJ (z) is the transmission coefficient relative to the J-th soliton. Repeating the argument, we find φn (zj ) ∼

zjn

J Y

l=j+1

  1 al (zj ) 0

nj+1  n  nj .

Upon passing through the j-th soliton, since the corresponding state is a bound state, we get φn (zj ) ∼

zj−n Sj

J Y

l=j+1

  0 al (zj ) 1

55

nj  n  nj−1 .

(3.195)

On the other hand, starting from n  n1 and proceeding in a similar way we find for the Jost function ψn the following asymptotic behavior ψn (zj ) ∼

zj−n

j−1 Y

zl−2 z¯l2

l=1

  0 al (zj ) 1




nj  n  nj−1

(3.196)

where we have used (3.78) and (3.80) in order to get the transmission coefficient for the right eigenfunctions, as well as the result (3.149). Comparing (3.195) and (3.196) and recalling (3.98), we get Sj

J Y

j−1 Y

al (zj ) = bj

al (zj ).

(3.197)

l=1

l=j+1

It is convenient to write Sj as −1 2

Sj (τ ) = ei(zj −zj

) τ +2δj +iψj

so that (3.98) yields −1 2

bj (τ ) ∼ ei(zj −zj

) τ +2δj− +iψj−

J Y

al (zj )

j−1 Y

m=1

l=j+1


−2 2 zm z¯m am (zj )−1

τ → −∞

where δj− , ψj− denote the asymptotics of the real functions δj , ψj as τ → −∞. Proceeding in a similar fashion as τ → +∞ and taking into account that the order of solitons is reversed, we get −1 2

bj (τ ) ∼ ei(zj −zj

) τ +2δj+ +iψj+

j−1 Y

al (zj )

J Y

m=j+1

l=1


−2 2 zm z¯m am (zj )−1

τ → +∞

therefore we conclude that +

−

e2(δj −δj

)+i(ψj+ −ψj− )

=

j−1 Y

al (zj )2

J Y

m=j+1

l=1


−2 2 zm z¯m am (zj )−2

or, explicitly, using the formula (3.119) for the pure 1-soliton transmission coefficient !2 J ! j−1 2 2 2 ∗ −2 2 Y Y
− z z z − (z ) + − + − m j j l −2 2 e2(δj −δj )+i(ψj −ψj ) = . zm z¯m 2 zj2 − (zl∗ )−2 zj2 − zm m=j+1

(3.198)

l=1

According to (3.189), this last formula provides the phase shift of the j-th soliton on the transition between the asymptotic states t → ±∞ due to the interaction with the others. For instance, in the 2-soliton case we have z 2 − z¯2
z 2 − z¯2

1 2 1 2 + − + − δ1 − δ1 = − δ2 − δ2 = log 2 (3.199) (z1 − z22 ) (¯ z12 − z¯22 ) "
2
# 2 2 2 z − z ¯ z ¯ − z ¯ 1 2 1 2 (3.200) ψ1+ − ψ1− = arg z22 z¯2−2 2 (z1 − z22 ) (¯ z12 − z22 ) Note that letting z1 = e−ihk1 , z2 = e−ihk2 , in the continuous limit (i.e. for h → 0) these expressions give back (2.114)–(2.115). 56

3.3

Conserved quantities and Hamiltonian structure

We showed that the scattering coefficient a(z) is time-independent. Since a(z) is analytic for |z| > 1, it admits a Laurent expansion whose coefficients are constants of the motion, as well. From the representation (3.88) for a(z), it follows that the quantities +∞ X

Qn Mn(2),−2j+1

(3.201)

n=−∞ (2),−j

are conserved for any integer j ≥ 0 and the coefficients Mn of the asymptotic expansion of (2) Mn can be calculated iteratively from (3.62)–(3.63). For instance, the first two constants of the motion are given by Γ1 =

+∞ X

Qn Rn−1 ,

+∞  X

Γ2 =

n=−∞

Qn Rn−2 −

n=−∞

1 2 (Qn Rn−1 ) 2



(3.202)

where we used the first one to simplify the expression of the second one which is obtained directly from (3.201) for j = 2. The scattering coefficient a ¯(z) is also a constant of the motion and proceeding exactly as before one can obtain a second set of conserved quantities given by +∞ X

¯ (1),2j−1 Rn M n

(3.203)

n=−∞

for any j ≥ 1, giving ¯1 = Γ

+∞ X

Qn−1 Rn ,

¯2 = Γ

n=−∞

 +∞  X 1 2 Qn−2 Rn − (Qn−1 Rn ) . 2 n=−∞

(3.204)

Note also that, taking into account the τ -dependence of the scattering coefficients (3.182), the determinant of the scattering matrix is a constant of the motion, i.e.     a(z, 0) ¯b(z, 0) a(z, τ ) ¯b(z, τ ) = det det b(z, 0) a ¯(z, 0) b(z, τ ) a ¯(z, τ ) and then, from (3.76) it follows that +∞ Y

c−∞ (τ ) = c−∞ (0) =

(1 − Rn Qn ) .

(3.205)

n=−∞

Note that when the potentials satisfy the symmetry condition Rn = ∓Q∗n , the first constants of the motion in (3.202) and (3.204) become Γ1 = ∓

+∞ X

Qn Q∗n−1 ,

¯ 1 = Γ∗ . Γ 1

(3.206)

n=−∞

Moreover, from (3.205) it also follows that c−∞ =

+∞ Y

n=−∞



57

2

1 ± |Qn |



(3.207)

is also a conserved quantity. The system of equations (3.5)–(3.6) is Hamiltonian [35] with the identification coordinates (q) : momenta (p) : Hamiltonian (H) :

Qn (τ )

(3.208)

Rn (τ ) +∞ +∞ X X log (1 − Rn Qn ) − Rn (Qn+1 + Qn−1 ) − 2

(3.209)

n=−∞

(3.210)

n=−∞

with the noncanonical (i.e. non constant) brackets {Qm (τ ), Rn (τ )} = i (1 − Rn (τ )Qn (τ )) δn,m {Qn (τ ), Qm (τ )} = {Qn (τ ), Rm (τ )} = 0.

(3.211) (3.212)

Unlike what happens in the continuous case (cf. (2.136)), since the bracket (3.211) is not constant, the Jacobi identity (2.128) does not follow trivially from skew-symmetricity but has to be checked ¯ 1 and c−∞ in separately. Note that the Hamiltonian is given by the conserved quantities Γ1 , Γ (3.202), (3.204) and (3.205). In the case when the initial data satisfy the additional constraints Rn = ∓Q∗n , we identify coordinates Qn and momenta Q∗n . The corresponding Hamiltonian and brackets are given by H=±

+∞ X

Q∗n (Qn+1 + Qn−1 ) ± 2

n=−∞

+∞ X

n=−∞

  2 log 1 ± |Qn |

  2 {Qm , Q∗n } = i 1 ± |Qn | δn,m {Qn , Qm } =

{Q∗n , Q∗m }

58

= 0.

(3.213)

(3.214) (3.215)

Chapter 4

Vector Nonlinear Schr¨ odinger equation 4.1

The inverse scattering transform for VNLS

The Lax pair for the vector nonlinear Schr¨odinger equation (VNLS) (1.3) is naturally obtained from the matrix generalization of the linear system (2.1)–(2.2), i.e.   −ikIN q v (4.1) vx = r ikIM and vt =

2ik 2 IN + iqr −2kq − iqx −2kr + irx −2ik 2 IM − irq

!

v

(4.2)

where v is an M -component vector, q = q(x, t) is an N × M matrix, r = r(x, t) is an M × N matrix and IN , IM are N ×N and M ×M identity matrices, respectively. Indeed, the compatibility condition (i.e. the equality of the mixed derivatives vxt = vtx ) is equivalent to the statement that q and r satisfy the evolution equations iqt = qxx − 2qrq

(4.3)

−irt = rxx − 2rqr

(4.4)

if k, the scattering parameter, is independent of x and t. Under the reduction r = ∓qH the system (4.3)–(4.4) corresponds to the single PDE iqt = qxx ± 2qqH q

(4.5)

and if q is either a row vector (N = 1) or a column vector (M = 1) it gives the VNLS (1.3). As usual, we refer to the equation with the x derivative, eq. (4.1), as the scattering problem and the equation with the t derivative, eq. (4.2), as the time dependence. The Inverse Scattering Transform for the two-component VNLS was first studied in [12]. Here we develop the IST on the infinite line for the more general matrix system (4.3)–(4.4) and then consider the reduction r = ∓qH as a special case. The treatment of the direct problem given here follows [1],[3]. The inverse problem is formulated as a Riemann-Hilbert boundary value problem, following [3]. The Gel’fand-Levitan-Marchenko integral equations follow from the RH problem. 59

4.1.1

Direct scattering problem

Jost functions and integral equations We refer to solutions of the scattering problem (4.1) as eigenfunctions with respect to the parameter k. When the potentials q, r → 0 rapidly as x → ±∞, the eigenfunctions are asymptotic to the solutions of   −ikIN 0 vx = v 0 ikIM when |x| is large. The solutions of this differential equation have the bases     IN 0 ¯ k) ∼ φ(x, k) ∼ eikx as x → −∞ e−ikx , φ(x, 0 IM ψ(x, k) ∼



0 IM



eikx ,

¯ k) ∼ ψ(x,



IN 0



e−ikx

as x → +∞

(4.6)

(4.7)

which are matrix valued functions with the following dimensions φ(x, k) :

(N + M ) × N,

ψ(x, k) :

(N + M ) × M,

¯ k) : φ(x, ¯ k) : ψ(x,

(N + M ) × M (N + M ) × N.

In the following analysis, it is convenient to consider functions with constant boundary conditions. Hence, we define the Jost functions as follows ¯ ¯ k), M(x, k) = e−ikx φ(x, ¯ k). ¯ N(x, k) = eikx ψ(x,

M(x, k) = eikx φ(x, k), N(x, k) = e−ikx ψ(x, k),

(4.8)

If the scattering problem (4.1) is rewritten as vx = (ikJ + Q) v

(4.9)

where J=



−IN 0 0 IM



,

Q=



0q r0



(4.10)

¯ then the Jost functions M(x, k) and N(x, k) are solutions of the differential equation χx (x, k) = ik(J + IN +M )χ(x, k) + (Qχ) (x, k)

(4.11)

¯ while N(x, k) and M(x, k) satisfies χ ˜x (x, k) = ik(J − IN +M )χ(x, ˜ k) + (Qχ) ˜ (x, k) with the constant boundary conditions   IN , M(x, k) ∼ 0   0 N(x, k) ∼ , IM

¯ M(x, k) ∼ ¯ N(x, k) ∼ 60





0 IM IN 0





(4.12)

as x → −∞

(4.13)

as x → +∞.

(4.14)

It is convenient to introduce the following notation: an (N + M ) × J matrix A will be denoted as  (up)  A A= A(dn) where the superscripts up and dn indicate, respectively, the first N columns and the last M columns of matrix A (i.e. A(up) is the N × J upper block and A(dn) is the lower M × J block of matrix A). Solutions of the differential equations (4.11)–(4.12) can be represented by means of the following integral equations Z +∞ G(x − ξ, k) (Qχ) (ξ, k)dξ χ(x, k) = w + −∞ +∞

χ(x, ˜ k) = w ˜+

Z

˜ − ξ, k) (Qχ) G(x ˜ (ξ, k)dξ

−∞

or, in component form, χm,j (x, k) = wm,j + χ ˜m,j (x, k) = w ˜ m,j +

Z Z

+M +∞ NX

−∞

+M +∞ NX

−∞

Gm,` (x − ξ, k) (Qχ)`,j (ξ, k)dξ

m = 1, . . . , N + M,

j = 1, . . . , N

˜ m,` (x − ξ, k) (Qχ) G ˜ `,j (ξ, k)dξ

m = 1, . . . , N + M,

j = 1, . . . , M

`=1

`=1

where w=



w(up) 0



,

w ˜ =



0 w ˜ (dn)



  ˜ ˜ j` (x, k) and the Green’s functions G(x, k) = (Gj` (x, k))j,`=1,2 and G(x, k) = G

j,`=1,2

are (N +

M ) × (N + M ) matrices satisfying the differential equations

˜ L− 0 G(x, k) = δ(x)IN +M

L+ 0 G(x, k) = δ(x)IN +M , L± 0 = I

N +M ∂x

− ik(J ± IN +M ).

The Green’s functions are not unique, and, as we show below, the choice of the Green’s function and the choice of the inhomogeneous terms w and w ˜ together determine the Jost function and its analytic properties. In analogy with the scalar case, we use the Fourier transform method and find  Z  −1 1 p IN 0 eipx dp G± (x, k) = 0 (p − 2k)−1 IM 2πi C±  Z  (p + 2k)−1 IN 0 ˜ ± (x, k) = 1 G eipx dp 0 p−1 IM 2πi C± where C± are the contours from −∞ to +∞ which, respectively, pass below and above both the singularities at p = 0 and p = 2k (see Fig. 2.1). Therefore we have    −2ikx  I 0 IN 0 ˜ ± (x, k) = ∓θ(∓x) e G± (x, k) = ±θ(±x) N 2ikx , G . (4.15) 0 e IM 0 IM

61

The “+” functions are analytic in the upper half plane of k and the “−” functions are analytic in the lower half plane. Taking into account the boundary conditions (4.13)–(4.14), we get the following integral equations for the Jost solutions   Z +∞ IN + G+ (x − ξ, k) (QM) (ξ, k)dξ (4.16) M(x, k) = 0 −∞   Z +∞ 0 ˜ + (x − ξ, k) (QN) (ξ, k)dξ N(x, k) = G (4.17) + IM −∞   Z +∞
0 ¯ ˜ − (x − ξ, k) QM ¯ (ξ, k)dξ M(x, k) = + G (4.18) IM −∞   Z +∞
IN ¯ ¯ (ξ, k)dξ. N(x, k) = G− (x − ξ, k) QN (4.19) + 0 −∞

Eqs. (4.16)–(4.19) are Volterra integral equations. The results of Lemma 2.1 can be generalized to the matrix case to show that if q,r ∈ L1 (R) with respect to any matrix norm, i.e. Z +∞ kqk1 = kqka (ξ)dξ < +∞ −∞ +∞

krk1 =

Z

−∞

krka (ξ)dξ < +∞

where k·ka is any matrix norm, then the Neumann series of the integral equations for M and N converge absolutely and uniformly (in x and k) in the upper half k-plane, while the Neumann ¯ and N ¯ converge absolutely and uniformly (in x and k) in the series of the integral equations for M lower half k-plane. This fact immediately implies that M(x, k) and N(x, k) are analytic functions ¯ ¯ of k for Imk > 0 and continuous for Im k ≥ 0 and M(x, k), N(x, k) are analytic functions of k for Imk < 0 and continuous for Im k ≤ 0. Lemma 4.1 If q, r ∈ L1 (R) then M(x, k), N(x, k) defined by (4.16)–(4.17) are analytic functions ¯ ¯ of k for Im k > 0 and continuous for Im k ≥ 0, while M(x, k), N(x, k) defined by (4.18)–(4.19) are analytic functions of k for Im k < 0 and continuous for Im k ≤ 0. Proof 4.1 We will prove the result for M(x, k). Analogously it can be proved for the remaining eigenfunctions. The Neumann series ∞ X M(x, k) = C(j) (x, k) (4.20) j=0

where C(0) (x, k) = C(j+1) (x, k) =



Z

IN 0



+∞

−∞

  G+ (x − ξ, k) QC(j) (ξ, k)dξ

j≥0

is, formally, a solution of the integral equation (4.16). In upper/lower component form Z x (j+1),up C (x, k) = q(ξ)C(j),dn (ξ, k)dξ −∞ Z x (j+1),dn C (x, k) = e2ik(x−ξ) r(ξ)C(j),up (ξ, k)dξ. −∞

62

(4.21)

Because C(0),dn = 0 we have for any j ≥ 0 C(2j+1),up = 0,

C(2j),dn = 0.

Using the identities 1 j!

Z

x

|f (ξ)|

−∞

"Z

ξ

−∞

0

|f (ξ )| dξ

0

#j

1 dξ = (j + 1)! =

1 (j + 1)!

Z

x

−∞

Z

d dξ

"Z

ξ

|f (ξ )| dξ

−∞

x

−∞

0

|f (ξ)| dξ

0

#j+1

dξ

j+1

where f ∈ L1 (R), one can show by induction that for Im k ≥ 0 R j+1 j R x x

krk (ξ)dξ kqk (ξ)dξ a a −∞ −∞

(2j+1),dn

C

(x, k) ≤ j! (j + 1)! a j j R R x x

krk (ξ)dξ kqk (ξ)dξ a a −∞ −∞

(2j),up .

C

(x, k) ≤ j! j! a

Therefore, if q, r ∈ L1 (R) the series (4.20) is majorized in norm by a uniformly convergent power series, which proves that the Neumann series (4.20) is itself uniformly convergent for Im k ≥ 0. From the integral equations (4.16)–(4.19) we may compute the asymptotic expansion for large k of the Jost functions. Integration by parts yields ! Rx 1 q(ξ)r(ξ)dξ IN − 2ik −∞ + O(k −2 ) (4.22) M(x, k) = 1 − 2ik r(x) ! R +∞ 1 IN + 2ik q(ξ)r(ξ)dξ x ¯ N(x, k) = + O(k −2 ) (4.23) 1 r(x) − 2ik ! 1 2ik q(x) N(x, k) = + O(k −2 ) (4.24) R +∞ 1 IM − 2ik r(ξ)q(ξ)dξ x ! 1 q(x) 2ik ¯ Rx M(x, k) = + O(k −2 ). (4.25) 1 IM + 2ik r(ξ)q(ξ)dξ −∞

Scattering Data

The two eigenfunctions with fixed boundary conditions as x → −∞ are linearly independent, as are the two eigenfunctions with fixed boundary conditions as x → +∞. Indeed, for any system of differential equations dv = Av dx
where A = (Aij )i,j=1,...,n and v(x) = v (1) (x), . . . , v (n) (x) , the Wronskian of the set of solutions v1 , . . . , vn defined as   (1) (n) v1 . . . v1   W (v1 , . . . , vn ) = det  . . . . . . . . .  (1) (n) vn . . . vn 63

satisfies the differential equation d W (v1 , . . . , vn ) = trA W (v1 , . . . , vn ) . dx Therefore, in if u(x, k) and v(x, k) are any two solutions of (4.1), and the Wronskian of u and v is given by W (u(x, k), v(x, k)) = det (u(x, k), v(x, k)) ,

(4.26)

d W (u, v) = ik (M − N ) W (u, v). dx

(4.27)

it satisfies

Taking into account the asymptotics (4.6)-(4.7), (4.27) yields
¯ k) = eik(M−N )x W φ(x, k), φ(x,
¯ k) (x, k) = −eik(M−N )x W ψ(x, k), ψ(x,

(4.28) (4.29)

¯ are linearly independent, and so are ψ and ψ. ¯ Therefore which proves that the functions φ, φ ¯ ¯ can write φ(x, k) and φ(x, k) as linear combinations of ψ(x, k) and ψ(x, k) or vice-versa. The coefficients of these linear combinations depend on k. Hence, we can write ¯ k)a(k) φ(x, k) = ψ(x, k)b(k) + ψ(x, ¯ k) = ψ(x, k)¯ ¯ k)b(k) ¯ φ(x, a(k) + ψ(x,

(4.30) (4.31)

where a(k) and ¯ a(k) are square matrices, N × N and M × M respectively, while b(k) is an M × N ¯ matrix and b(k) is an N × M matrix. The relations (4.30)–(4.31) hold for any k such that all four eigenfunctions exist. In particular, they hold for Im k = 0 and define the scattering matrix   ¯ a(ξ) b(ξ) S(ξ) = ξ∈R b(ξ) ¯ a(ξ) in terms of which (4.30)–(4.31) can be written as

¯ ξ) = ψ(x, ¯ ξ), ψ(x, ξ) S(ξ). φ(x, ξ), φ(x,
¯ as x → ±∞ with eqs. (4.30)–(4.31) shows that the Comparing the asymptotics of W φ, φ scattering matrix is unimodular, i.e.   ¯ a(ξ) b(ξ) det S(ξ) = det =1 ξ ∈ R. (4.32) b(ξ) ¯ a(ξ) ¯ as linear It also convenient to introduce the “right” scattering coefficients expressing ψ and ψ ¯ i.e. combinations of φ and φ, ¯ k)c(k) ψ(x, k) = φ(x, k)d(k) + φ(x, ¯ ¯ ¯ ψ(x, k) = φ(x, k)¯ c(k) + φ(x, k)d(k).

(4.33) (4.34)

The scattering coefficients can be related to the Wronskian of the Jost solutions in the following way     
a(k)
0 ¯ ¯ W (φ(x, k), ψ(x, k)) = W ψ(x, k), ψ(x, k) , ψ(x, k), ψ(x, k) b(k) IM   
a(k) 0 ¯ k), ψ(x, k) ¯ k), ψ(x, k)) det a(k) =W ψ(x, = W (ψ(x, (4.35) b(k) IM 64

and similarly ¯ k), φ(x, ¯ k)) = W (ψ(x, ¯ k), ψ(x, k)) det ¯ W (ψ(x, a(k).

(4.36)

Form the other side, using the “right” data we get the relations ¯ k)) W (φ(x, k), ψ(x, k)) = det c(k)W (φ(x, k), φ(x, ¯ k), φ(x, ¯ k)) = det ¯ ¯ k)) W (ψ(x, c(k)W (φ(x, k), φ(x,

(4.37) (4.38)

and the comparison between (4.35)–(4.36) and (4.37)–(4.38) yields det a(k) = det c(k)

(4.39)

det ¯ a(k) = det ¯ c(k).

(4.40)

One can also derive the following integral relationships for the scattering coefficients Z +∞ a(k) = IN + q(ξ)M(dn) (ξ, k)dξ,

(4.41)

−∞

b(k) =

Z

+∞

e−2ikξ r(ξ)M(up) (ξ, k)dξ,

(4.42)

−∞

a(k) = IM + ¯

Z

+∞

¯ (up) (ξ, k)dξ, r(ξ)M

(4.43)

−∞

¯ b(k) =

Z

+∞

¯ (dn) (ξ, k)dξ, e2ikξ q(ξ)M

(4.44)

−∞

and ¯ c(k) = IN − ¯ d(k) =−

Z

Z

+∞

¯ (dn) (ξ, k)dξ q(ξ)N

−∞ +∞ −2ikξ

e

(4.45)

¯ (up) (ξ, k)dξ r(ξ)N

(4.46)

−∞

c(k) = IM − d(k) = −

Z

Z

+∞

r(ξ)N(up) (ξ, k)dξ

−∞ +∞ 2ikξ

e

(4.47)

q(ξ)N(dn) (ξ, k)dξ.

(4.48)

−∞

Indeed, introducing ¯ ∆(x, k) = M(x, k) − N(x, k)a(k). ¯ (4.16), (4.19), as well as the relation and using the integral equations for M and N   IN 0 G+ (x, k) − G− (x, k) = , 0 e2ikx IM we can write ∆(x, k)−

Z

+∞

G− (x − ξ, k) (Q∆) (ξ, k)dξ

−∞

=



IN − a(k) 0



−

Z

+∞ −∞

65



q(ξ)M(dn) (ξ, k) 2ik(x−ξ) e r(ξ)M(up) (ξ, k)



dξ.

(4.49)

From the other side, the scattering equation (4.30) yields ∆(x, k) = e2ikx N(x, k)b(k) and then ∆(x, k) −

Z

+∞

G− (x − ξ, k) (Q∆) (ξ, k) dξ = −∞



0 b(k)



e2ikx

(4.50)

where we used the integral equation (4.17) for N(x, k) as well as the identity ˜ + (x, k). G− (x, k) = e2ikx G Comparing (4.49) and (4.50) we get the integral representations (4.41) and (4.42). Eqs. (4.43) and ¯ ¯ (4.44) are derived analogously, by considering ∆(x, k) = M(x, k) − N(x, k)¯ a(k). Similarly, one obtains the integral representations for the “right” data (4.45)–(4.48). ¯ Since M(x, k) and M(x, k) are analytic, respectively, for Im k > 0 and Im k < 0, and continuous on Im k = 0, the integral representations (4.41) and (4.43) immediately yield that a(k) is analytic in the UHP and continuous for Im k = 0 and ¯ a(k) is analytic in the LHP and continuous for ¯ Im k = 0; b(k) and b(k) cannot in general be continued off the real axis. From the integral representations (4.41) and (4.43) and the asymptotics (4.24), (4.25) it also follows that Z +∞ 1 q(ξ)r(ξ)dξ + O(k −2 ) (4.51) a(k) = IN − 2ik −∞ Z +∞ 1 a(k) = IM + ¯ r(ξ)q(ξ)dξ + O(k −2 ). (4.52) 2ik −∞ and, under the assumption det a(ξ) 6= 0, det ¯ a(ξ) 6= 0 for all ξ ∈ R, these asymptotics, together with the analytic properties we have estabilished, yield that the zeros of det a(k) in the UHP and the zeros of det ¯ a(k) are in finite number (no cluster points on Im k = 0 are allowed since a(k) and ¯ a(k) are continuous on the real k-axis). Note that the eqs.(4.30) and (4.31) can be written as ¯ µ(x, k) = N(x, k) + e2ikx N(x, k)ρ(k) ¯ µ ¯ (x, k) = N(x, k) + e−2ikx N(x, k)ρ(k) ¯

(4.53) (4.54)

where we introduced µ(x, k) = M(x, k)a−1 (k),

¯ µ ¯ (x, k) = M(x, k)¯ a−1 (k)

(4.55)

¯ a−1 (k). ρ(k) ¯ = b(k)¯

(4.56)

and the reflection coefficients ρ(k) = b(k)a−1 (k),

Moreover, from (4.30)–(4.31) and (4.33)–(4.34) it follows that for any ξ ∈ R   −1  ¯ ¯ c(ξ) d(ξ) a(ξ) b(ξ) = ¯ d(ξ) c(ξ) b(ξ) ¯ a(ξ) and therefore, under the assumptions det a(ξ) 6= 0, det ¯ a (ξ) 6= 0 for all ξ ∈ R, the following relations between “left” and “right” scattering data hold: (IN − ρ(ξ)ρ(ξ)) ¯ a(ξ)¯ c (ξ) = IN

(4.57)

(IM − ρ(ξ) ρ(ξ)) ¯ a(ξ)c (ξ) = IM . ¯

(4.58)

66

Proper Eigenvalues and Norming Constants As usual, we define a proper eigenvalue to be a (complex) value of k for which the scattering problem (4.1) admits a bounded solution that decays as x → ±∞. If Im k > 0, from the asymptotics (4.6)–(4.7) it follows that φ(x, k) decays as x → −∞ while ψ(x, k) decays as x → +∞. Therefore, in order for kj = ξj + iκj in the UHP being an eigenvalue, it must be that one of the solutions is in the span of the other one, i.e. W (φ(x, kj ), ψ(x, kj )) = 0. Similarly, k¯j in the LHP is an eigenvalue if, and only if, ¯ kj ), ψ(x, ¯ kj )) = 0. W (φ(x, From the other side, eqs. (4.35)–(4.36) show that the eigenvalues in the UHP are the points k = kj such that det a(kj ) = 0 and the eigenvalues in the LHP are the zeros k¯j of det ¯ a(k). There are no eigenvalues on the real k-axis because none of the basis eigenfunctions vanishes as x → ±∞ if Im k = 0, and we also assume that det a(ξ) 6= 0, det ¯ a(ξ) 6= 0 for any ξ ∈ R. From eqs. (4.55) it follows that µ(x, k) is a meromorphic function with poles in correspondence of the zeros of det a(k) and µ ¯ (x, k) is a meromorphic function with poles at the zeros of det ¯ a(k). J Let us assume that det a(k) has J simple zeros at the points {kj = ξj + iκj : κj > 0}j=1 and  J¯ det a ¯(k) has J¯ simple zeros at k¯` = ξ¯` − i¯ κ` : κ ¯` > 0 `=1 . Let α(k) be the cofactor matrix of a(k) and let us introduce the notation a(k) = det a(k). Then from eq. (4.53) it follows Res (µ ; kj ) = lim (k − kj ) µ(x, k) k→kj

= e2ikj x N (x, kj )

1 a0 (kj )

b(kj )α(kj )

= e2ikj x N (x, kj ) Cj

(4.59)

where 0 denotes derivative (with respect to the parameter k). For any j = 1, . . . , J Cj =

1 b(kj )α(kj ) a0 (kj )

(4.60)

is an M × N matrix and we refer to it as the norming constant associated to the discrete eigenvalue kj . By a similar procedure, we can find the expression for the residues of the poles of µ ¯ (x, k)

where for any ` = 1, . . . , J¯



¯ ¯ x, k¯` C ¯` Res µ ¯ ; k¯` = e−2ik` x N ¯` = C

1 ¯ ¯ b(k` )α( ¯ k¯` ) a ¯0 (k¯` )

(4.61)

(4.62)

is an N × M matrix (as before, a ¯(k) = det ¯ a(k) and α(k) ¯ is the cofactor matrix of ¯ a(k)).  J¯ J To summarize, in the general case the eigenvalues {kj : Im kj > 0}j=1 ∪ k¯` : Im k¯` < 0 `=1  J¯ J ¯` and the associated norming constants {Cj }j=1 ∪ C , together with the reflection coefficients `=1 {ρ(ξ), ρ(ξ) ¯ : ξ ∈ R} given by (4.56) constitute the set of the scattering data.

67

Symmetry Reductions The M -component VNLS equation is a special case of the system (4.3)–(4.4) where r = ∓qH and q is an M -component row vector (i.e. N = 1). More generally, we can consider the symmetry r = ∓qH

(4.63)

for any q. Note that when r = qH , the scattering problem (4.1) with potentials decaying rapidly enough as x → ±∞ does not admit discrete eigenvalues off the real k-axis. The symmetries (4.63) in the potentials induce symmetries in the scattering data. Indeed, let us consider the matrix-valued functions defined in the upper half k-plane

where

  ¯ k ∗ ) H φ(x, k) f± (x, k) = σ ± φ(x, σ± =



IN 0 0 ±IM



.

(4.64)

From the scattering problem (4.1) it follows that under the symmetry (4.63) ∂ f± (x, k) = 0 ∂x and equating the asymptotic behavior of f± as x → +∞ and x → −∞ following from (4.6)–(4.7) and (4.30)–(4.31), we find that ¯H (k ∗ )a(k) = 0 ¯H (k ∗ )b(k) ± b a or ρ¯H (k ∗ ) = ∓ρ(k)

(4.65)

  ¯ k ∗ ) H ψ(x, k) and for any k ∈ R. An analogous argument yields that g± (x, k) = σ ± φ(x,   ¯ (x, k ∗ ) H φ(x, k), defined in the upper half k-plane, are independent of x and h± (x, k) = σ ± ψ comparing the asymptotics of g± and h± as |x| → ∞ following from (4.6)–(4.7) and (4.33)–(4.34) we get ¯H (k ∗ ) = c(k) a ¯ cH (k ∗ ) = a(k).

(4.66) (4.67)

det c(k) = (det ¯ a(k ∗ ))∗ .

(4.68)

In particular, we have

Taking into account (4.39)–(4.40), under the reduction (4.63) we also have det ¯ a(k ∗ ) = (det a(k))

∗

(4.69)

which implies that kj in the UHP is an eigenvalue if, and only if, k¯j = kj∗ 68

(4.70)

¯ i.e. the number of eigenvalues in the UHP is is an eigenvalue in the LHP. Consequently J = J, equal to the number of eigenvalues in the LHP. As far as the associated norming constants are concerned, from (4.65) it follows ¯ j = ∓CH C j .

(4.71)

  ¯ (x, k ∗ ) H φ(x, ¯ k) and ˆ Moreover, if we consider the matrix-valued functions ˆ f± (x, k) = σ ± φ g± (x, k) = H ∗ [σ ± φ (x, k )] φ(x, k) defind on the real k-axis, the comparison of their asymptotic behaviors as x → ±∞ yields the following characterization equations for the scattering data aH (k ∗ ) a(k) = IN ∓ bH (k ∗ ) b(k) ¯H (k ∗ ) b(k) ¯ aH (k ∗ ) ¯ ¯ a(k) = IM ∓ b

Im k = 0 Im k = 0

(4.72) (4.73)

which can be considered as defining a matrix Riemann-Hilbert problem with zeros (matrix factorization problem) for a(k) and ¯ a(k) with jump given accross the real axis and boundary conditions given by (4.51)–(4.52). Trace formula J

Assume a(k) = det a(k) and a ¯(k) = det ¯ a(k) to have the simple zeros {kj : Im kj > 0}j=1 and  J¯ k¯j : Im k¯j < 0 , respectively, and define j=1

α(k) =

J ∗ Y k − km a(k), k − km m=1

α ¯ (k) =

J¯ ∗ Y k − k¯m a ¯(k). ¯ k − km m=1

(4.74)

α(k) is analytic in the UHP, whereas it has no zeros, while α ¯ (k) is analytic in the LHP, whereas it has no zeros; moreover, due to (4.51)–(4.52), α(k), α(k) ¯ → 1 as |k| → ∞ in the proper half plane. Therefore we have   Z +∞ J X k − km 1 log (α(ξ)¯ α(ξ)) dξ, Im k > 0 (4.75) log a(k) = log + ∗ k − km 2πi −∞ ξ−k m=1   Z +∞ J X k − k¯m 1 log (α(ξ)¯ α(ξ)) log dξ Im k < 0. (4.76) log a ¯(k) = − ¯∗ 2πi ξ − k k − k −∞ m m=1 If the symmetry (4.63) holds, from (4.69) and (4.71) it follows that α(ξ)¯ α(ξ) = a(ξ)¯ a(ξ) = |a(ξ)|2 2 −1 and using (4.57) |a(ξ)| = det (IN − ρ(ξ)ρ(ξ)) ¯ . Hence, eq. (4.75) allows one to recover a(k), a ¯(k)  J¯ J ¯ ¯ for any k in the proper half plane from knowledge of {kj , Im kj > 0}j=1 , kj , Im kj > 0 j=1 and
−1 α(ξ)¯ α(ξ) = det IN ± ρH (ξ) ρ(ξ) for ξ ∈ R. The problem of reconstructing the matrices a(k) and ¯ a(k) is more complicated. We do note, however, that from from (4.41) and (4.43) and the inverse scattering below, the (matrix) scattering coefficients a(k) and ¯ a(k) can be reconstructed, in principle, from the scattering data  J¯ J ¯ ¯ {kj , Cj } , kj , Cj and {ρ(ξ), ρ(ξ) ¯ : ξ ∈ R}. j=1

4.1.2

j=1

Inverse Scattering

The inverse problem consists of constructing a map from the scattering data back to the potentials. J We start with (i) the reflection coefficients ρ(ξ) and ρ(ξ) ¯ for ξ ∈ R (ii) the eigenvalues {kj }j=1 69

 J¯  J¯ J ¯j and k¯j j=1 and (iii) the corresponding norming constants {Cj }j=1 and C . First we use j=1 these data to recover the Jost functions. Then, we recover the potentials in terms of these Jost functions. ¯ In the previous section, we showed that the functions N(x, k) and N(x, k) exist and are analytic in the regions Im k > 0 and Im k < 0 respectively, if q, r ∈ L1 (R). Similarly, under the same conditions on the potentials, the functions µ(x, k) and µ ¯ (x, k) defined by (4.55) are meromorphic in the regions Im k > 0 and Im k < 0 respectively. Therefore, in the inverse problem we assume the unknown functions are sectionally meromorphic. With this assumption, the equations (4.53)– (4.54) can be considered to be the jump conditions of a Riemann-Hilbert problem. To recover the sectionally meromorphic functions from the scattering data, we convert the Riemann-Hilbert problem to a system of linear integral equations. Case of no poles We begin by solving the Riemann-Hilbert problem in the case where µ and µ ¯ have no poles. Introducing the (N + M ) × (N + M ) matrices
¯ m+ (x, k) = (µ(x, k), N(x, k)) , m− (x, k) = N(x, k), µ(x, ¯ k) (4.77) the “jump” conditions (4.53)–(4.54) can be written as

m+ (x, k) − m− (x, k) = m− (x, k)V(x, k)

(4.78)

where V(x, k) =

−ρ(k)ρ(k) ¯

−2ikx −ρ(k)e ¯

ρ(k)e2ikx

0

!

(4.79)

and m± (x, k) → IN +M as |k| → ∞ in the proper half plane. Applying the projector P − defined in (2.66) to the eq. (4.78) yields Z +∞ 1 m− (x, ξ)V(x, ξ) m− (x, k) = IN +M + dξ (4.80) 2πi −∞ ξ − (k − i0) which allows one, in principle, to find m− (x, k). Note that as |k| → ∞ m− (x, k) = I−

1 2πik

Z

+∞

m− (x, ξ)V(x, ξ)dξ + O(k −2 )

(4.81)

−∞

and taking into account the asymptotics (4.22)–(4.25) and the definitions (4.78)–(4.79), eq. (4.81) gives the reconstruction of the potentials in terms of the scattering data, i.e. Z 1 +∞ 2ikx (dn) r(x) = e N (x, k)ρ(k)dk (4.82) π −∞ Z +∞ 1 ¯ (up) (x, k)ρ(k)dk. e−2ikx N ¯ (4.83) q(x) = π −∞

70

Case of poles Suppose now that the potentials are such that a(k) = det a(k) and a ¯(k) = det ¯ a(k) have a finite number of simple zeros in the regions Im k > 0 and Im k < 0, respectively, which we denote as  J¯ ¯(ξ) 6= 0 for any {kj , Im kj > 0}Jj=1 and k¯j , Im k¯j < 0 j=1 . We shall also assume that a(ξ) 6= 0, a − + ξ ∈ R. As before, we apply P to both sides of (4.53) and P to both sides of (4.54). Taking into account the analytic properties of the involved functions and the asymptotics (4.22)–(4.25), as well as (4.59), we obtain Z +∞ J X 1 e2iξx e2ikj x Nj (x)Cj + N(x, ξ)ρ(ξ)dξ (k − kj ) 2πi −∞ ξ − (k − i0) j=1

¯ N(x, k) =



IN 0



+

N(x, k) =



0 IM



Z +∞ J¯ ¯ X e−2ikj x ¯ 1 e−2iξx ¯ ¯ + N (x) C − N(x, ξ)¯ ρ(ξ)dξ j j ¯j ) 2πi ξ − (k − i0) (k − k −∞ j=1

(4.84)

(4.85)

¯ j (x) = N(x, ¯ where, as before Nj (x) = N(x, kj ), N k¯j ). In order to close the system, we evaluate ¯ ¯ eq. (4.84) at k = kl for any l = 1, . . . , J, and (4.85) at k = kj for any j = 1, . . . , J, thus getting ¯ l (x) = N



IN 0



+

Nj (x) =



0 IM



+

Z +∞ 2iξx J X 1 e2ikj x e N (x)C + N(x, ξ)ρ(ξ)dξ j j ¯ 2πi −∞ ξ − k¯` (k` − kj ) j=1

Z +∞ −2iξx J¯ ¯ X e−2ikm x ¯ 1 e ¯ ¯ N (x) C − ρ(ξ) ¯ N(x, ξ)dξ. m ¯m ) m 2πi (k − k j −∞ ξ − kj m=1

(4.86)

(4.87)

The equations (4.84)–(4.85) and (4.86)–(4.87) constitute a linear algebraic-integral system of equa¯ tions which, in principle, solve the inverse problem for the eigenfunctions N(x, k) and N(x, k). By comparing the asymptotic expansions at large k of the right-hand sides of (4.84) and (4.85) to the expansions (4.23) and (4.24), respectively, we obtain r(x) = −2i

J X

e

2ikj x

(dn) Nj (x)Cj

j=1

q(x) = 2i

J¯ X

¯

(up)

¯ e−2ikj x N j

1 + π

Z

+∞

1 π

Z

+∞

¯j + (x)C

j=1

e2iξx N(dn) (x, ξ)ρ(ξ)dξ

(4.88)

¯ (up) (x, ξ)ρ(ξ)dξ e−2iξx N ¯

(4.89)

−∞

−∞

which reconstruct the potentials and thus complete the formulation of the inverse problem. If the potentials decay rapidly enough at infinity, so that ρ(k) can be analytically continJ ued above all poles {kj , Im kj > 0}j=1 and ρ(k) ¯ can be analytically continued below all poles  J¯ , then the system of equations (4.84)–(4.85) and (4.86)–(4.87) can be simplified k¯j , Im k¯j < 0 j=1

as follows

¯ N(x, k) = N(x, k) =





IN 0 0 IM





1 + 2πi −

1 2πi

Z

C0

Z

¯0 C

e2iξx N(x, ξ)ρ(ξ)dξ ξ−k

(4.90)

e−2iξx ¯ N(x, ξ)ρ(ξ)dξ ¯ ξ−k

(4.91)

where C0 is a contour from −∞ to +∞ that passes above all zeros of a(k) and C¯0 is a contour from −∞ to +∞ that passes below all zeros of a ¯(k). In the same hypothesis, eqs. (4.88)–(4.89) 71

can be written as Z 1 r(x) = e2iξx N(dn) (x, ξ)ρ(ξ)dξ π C0 Z 1 ¯ (up) (x, ξ)ρ(ξ)dξ. e−2iξx N ¯ q(x) = π C¯0

(4.92) (4.93)

Gel’fand-Levitan-Marchenko equations We can also provide a reconstruction for the potentials by developing the Gel’fand-Levitan-Marchenko integral equation, instead of using the projection operators (cf. [1]). In analogy with the scalar case, we represent the eigenfunctions in terms of triangular kernels  Z +∞  0 + K(x, s)e−ik(x−s) ds s > x, Im k > 0 (4.94) N(x, k) = IM x   Z +∞ IN ¯ ¯ N(x, k) = + K(x, s)eik(x−s) ds s > x, Im k < 0 (4.95) 0 x ¯ are (N + M ) × M and (N + M ) × N matrices, respectively. where K and K R +∞ 1 −ik(x−y) for y > x to the equation (4.90), we get Applying the operator 2π −∞ dk e ¯ K(x, y) +



0 IM



+∞

Z

F(x + y) +

K(x, s)F(s + y)ds = 0

(4.96)

x

where F(x) =

1 2π

Z

ρ(ξ)eiξx dξ =

C0

Analogously, operating on eq. (4.91) with K(x, y) +



IN 0



1 2π

1 2π

+∞

Z

ρ(ξ)eiξx dξ − i

−∞

R +∞

Cj eikj x .

(4.97)

j=1

dk eik(x−y) for y > x gives

−∞

¯ + y) + F(x

J X

Z

+∞

¯ ¯ + y)ds = 0 K(x, s)F(s

(4.98)

x

where 1 ¯ F(x) = 2π

Z

C¯0

ρ(ξ)e ¯

−iξx

1 dξ = 2π

Z

+∞

ρ(ξ)e ¯

−∞

−iξx

dξ + i

J¯ X

¯ j e−ik¯j x . C

(4.99)

j=1

Eqs. (4.96) and (4.98) constitute the Gel’fand-Levitan-Marchenko integral equations. Inserting the representations (4.94)–(4.95) for the eigenfuctions into the eqs. (4.92)–(4.93) we obtain the reconstruction of the potentials in terms of the kernels of GLM equations, i.e. ¯ (up) (x, x). r(x) = −2K

q(x) = −2K(dn) (x, x),

(4.100)

If the symmetry (4.63) between the potentials holds, then, taking into account (4.65)–(4.71) we also have the symmetry ¯ F(x) = ∓FH (x).

72

(4.101)

4.1.3

Time evolution

The operator (4.2) fixing the evolution of the Jost functions can be written as   AB ∂t v = v CD

(4.102)

where B, C → 0 as x → ±∞ (since we have assumed that q, r → 0 as x → ±∞). Then the time dependence must asymptotically satisfy   A∞ IN 0 ∂t v = v as x → ±∞. (4.103) 0 −A∞ IM where we introduced A∞ = 2ik 2 such that lim A(x, k) = A∞ IN = 2ik 2 IN ,

|x|→∞

lim D(x, k) = −A∞ IM = −2ik 2 IM .

|x|→∞

The system (4.103) has solutions that are linear combinations of   A t   0 e ∞ IN − + , v = v = . 0 e−A∞ t IM However, such solutions are not compatible with the fixed boundary conditions of the Jost functions (4.6)–(4.7). We define time-dependent functions ¯ t) ¯ Φ(x, t) = e−A∞ t φ(x, A∞ t ¯ ¯ Ψ(x, t) = e ψ(x, t)

Φ(x, t) = eA∞ t φ(x, t), Ψ(x, t) = e−A∞ t ψ(x, t),

(4.104) (4.105)

to be solutions of the time-differential equation (4.102). Then the evolution equations for φ and ¯ become φ     B A − A∞ IN B ¯ ¯ = A + A∞ IN φ (4.106) φ, ∂t φ ∂t φ = C D + A∞ IM C D − A∞ IM so that, taking into account the eqs.(4.30)–(4.31) and evaluating (4.106) as x → +∞ one gets ∂t a(k) = 0,

∂t ¯ a(k) = 0 ¯ ¯ ∂t b(k) = 2A∞ b(k)

∂t b(k) = −2A∞ b(k), or explicitly a(k, t) = a(k, 0), b(k, t) = e

4ik2 t

a(k, t) = ¯ ¯ a(k, 0) 2 ¯ ¯ 0). b(k, 0), b(k, t) = e−4ik t b(k,

(4.107) (4.108)

From (4.107) it is clear that the eigenvalues (i.e. the zeros of a(k) and a ¯(k)) are constant as the solutions evolve. Not only the number of eigenvalues, but also their locations are fixed. Thus, the eigenvalues are time-independent discrete states of the evolution. On the other hand, the evolution of the reflection coefficients is given, like in the scalar case, by 2

ρ(k, t) = ρ(k, 0)e4ik t ,

ρ(k, ¯ t) = ρ(k, ¯ 0)e−4ik

2

t

(4.109)

and this gives also the evolution of the norming constants ¯ j (t) = C ¯ j (0)e−4ik¯j2 t . C

2

Cj (t) = Cj (0)e4ikj t , 73

(4.110)

4.2 4.2.1

Soliton Solutions One soliton solution

In the case where the scattering data comprise proper eigenvalues, but ρ(ξ) = ρ(ξ) ¯ = 0 for ξ ∈ R, the algebraic-integral system (4.84)–(4.85) and (4.86)–(4.87) reduces to the linear algebraic system ¯ l (x) = N



IN 0



+

Nj (x) =



0 IM



+

J X e2ikj x Nj (x)Cj ¯ (kl − kj ) j=1

J¯ ¯ X e−2ikm x ¯ ¯ m. Nm (x)C (kj − k¯m ) m=1

(4.111)

(4.112)

The one soliton solution is obtained for J = J¯ = 1. In the relevant physical case, when the potentials satisfy the symmetry r = −qH and N = 1, the matrix C1 reduces to an M -component ¯ 1 is an M -component row vector. Taking into account (4.70) and (4.71), column vector, while C we get ! IN 1 ¯ 2ik1 x N1 (x) = (4.113) 2 1k − e 2iη C1 −4ηx 1 + kC 4η 2 e !   ∗ ¯1 C 1 e−2ik1 x 0 2ik1 x (4.114) N1 (x) = + ¯1 IM 2iη 1 + kC12k2 e−4ηx − e 2iη C1 C 4η

where k1 = ξ + iη. From (4.89) it then follows that q(x) = −2iηe−2iξx sech(2ηx − 2δ)

CH 1 kC1 k

(4.115)

where e2δ =

kC1 k . 2η

(4.116)

Introducing the explicit time dependence of C1 as given by (4.110), one finally gets the one soliton solution of VNLS equation q(x, t) = 2η e−2iξx+4i(ξ

2

−η 2 )t−iπ/2

sech(2ηx − 8ξηt − 2δ0 ) p

(4.117)

where e2δ0 =

kSk , 2η

p=

S , kSk

S = CH 1 (0).

(4.118)

It is evident that the one soliton solution of VNLS is of the form q(x, t) = p q(x, t)

(4.119)

where q is the one soliton solution of scalar NLS (cf. (2.106)), thus an individual soliton of VNLS is fundamentally governed by the scalar NLS. However, the vector soliton (4.117) is also characterized by a polarization, i.e. the vector p, and the vector nature of the solution affects the dynamics when solitons with different polarization interact. 74

4.2.2

Transmission coefficients for the pure one soliton potential

In order to obtain the transmission coefficients relative to the pure one soliton solution (4.115), we insert (4.88)–(4.89) for J = J¯ = 1 and ρ(w) = ρ(w) ¯ = 0 into (4.45) and (4.47) to get Z +∞ 2i ¯ ¯ (up) (x)C ¯ 1 N(dn) (x)C1 dx e−2i(k1 −k1 )x N ¯ c(k) = IN − 1 1 k − k1 −∞ Z +∞ 2i ¯ (dn) ¯ (up) (x)C ¯ 1 dx. e2i(k1 −k1 )x N1 (x)C1 N c(k) = IM + 1 k − k¯1 −∞ For the sake of semplicity we now restrict to the case of the 2-component VNLS equation, i.e. we take N = 1, M = 2. Therefore ¯ c(k) reduces to a scalar and c(k) is a 2 × 2 matrix. Taking into account the symmetry (4.70)–(4.71) and substituting (4.113)–(4.114) we find c(k) = IM −

k1 − k1∗ 1 C1 CH 1 k − k1∗ kC1 k2

¯ c(k) = c¯(k) =

k − k1∗ . k − k1

(4.120) (4.121)

From the symmetries (4.66)-(4.67) it then follows k − k1 k − k1∗ k1 − k1∗ 1 C1 CH + 1 k − k1 kC1 k2

a(k) = a(k) =

(4.122)

¯ a(k) = IM

(4.123)

where, as before, a(k) reduces to a scalar (a(k)) and ¯ a(k) is a 2 × 2 matrix. In terms of the 2-component row vector S introduced in (4.118), eq. (4.123) can be written as ¯ a(k) = IM +

4.2.3

k1 − k1∗ 1 SH S k − k1 kSk2

(4.124)

Vector Symmetry

The vector equation (1.3) reduces to NLS (1.1) under the reduction q(x, t) = p q(x, t) where p = p(1) , . . . , p(N )


T

(4.125)

is a constant N -component vector such that 2

kpk = |p(1) |2 + · · · + |p(N ) |2 = 1

(4.126)

and q is a scalar function of x and t. As a consequence, any solution of scalar NLS generates a family of solutions of VNLS parametrized by the choice of p, to which we refer as reduction solutions. The vector p is usually called the polarization of the reduction solution. 2 The fact that the polarization of a reduction solution is arbitrary (up to the restriction kpk = 1) is a manifestation of a symmetry of VNLS. The dependent variable transformation q ˜ = Uq

(4.127)

where U is any unitary matrix, leaves VNLS invariant. Thus, if q is a reduction solution of the form (4.125), then also q ˜ is a reduction solution with p ˜ = Up and q˜ = q. The transformation (4.127) 75

can be interpreted as a change in the basis of the vector space of the dependent variable. For any reduction solution, the polarization depends entirely of the choice of the basis and by changing the basis via transformations of the form (4.127) one can arbitrarily change the polarization. We refer to the squared absolute value of a component of the polarization as the intensity of that component. By definition, the sum of the intensities, i.e. the total intensity, is one. However, the distribution of intensity among the components of the vector of any given reduction solution depends on the choice of the polarization basis.

4.2.4

Vector Soliton Interactions

Suppose we consider the solution of VNLS (r = −qH and N = 1) corresponding to the scattering data  kj = ξj + iηj , ηj > 0, Cj }Jj=1 ∪ {ρ(ξ) = 0 for ξ ∈ R .

In order to get the pure J-soliton solution, one can in principle solve the linear system (4.111)– (4.112). The problem of a J-soliton collision can be investigated by looking at the asymptotic states as t → ±∞ (cf. [12]). For a generic multiple-soliton solution of VNLS, with the individual solitons traveling at different velocities, in the backward (t → −∞) and forward (t → +∞) long-time limits, such a solution asymptotically breaks up into individual solitons q∼

J X

± p± j qj

t → ±∞

(4.128)

j=1

± − + where p± j is a complex unit vector and qj is the one soliton solution of NLS such that qj and qj are characterized by the same amplitude and velocity. Let us fix the values of the soliton parameters for t → −∞, i.e. for each eigenvalue kj = ξj + iηj we assign a vector S− j (see (4.117)–(4.118)) − that completely determines qj . For t → +∞ we denote the corresponding vectors by S+ j . For definiteness, let us assume that ξ1 < ξ2 < · · · < ξJ . Then, as t → −∞ the solitons are distributed along the x-axis in the order corrsponding to ξJ , ξJ−1 , . . . , ξ1 ; the order of the soliton sequence is reversed as t → +∞. In order to determine the result of the interaction between solitons, i.e. to − calculate S+ j given Sj , we trace the passage of the Jost functions through the asymptotic states [12]. We denote the soliton coordinates at the instant of time t by xj (t) (|t| is assumed large enough so that one can talk about individual solitons). If t → −∞ then xJ  xJ−1  · · ·  x1 . The function φ(x, kj ) has the form   IN −ikj x φ(x, kj ) ∼ e x  xJ . 0

After passing through the J-th soliton it will be of the form   IN aJ (kj ) φ(x, kj ) ∼ e−ikj x 0 aJ (k) is the transmission coefficient relative to the J-th soliton. Repeating the argument (see App. B for details), we find φ(x, kj ) ∼ e−ikj x



IN 0

 Y J

l=j+1 right

76

al (kj )

xj+1  x  xj

where the notation right indicates that the product is performed such that the matrix with index ` occurrs to the right of the matrix with index ` − 1, i.e. J Y

al (kj ) = aj+1 (kj )aj+2 (kj ) . . . aJ (kj ).

l=j+1 right

Upon passing through the j-th soliton, since the corresponding state is a bound state, we get φ(x, kj ) ∼ e

ikj x



0 IM



J Y

SH j

al (kj )

xj  x  xj−1 .

(4.129)

l=j+1 right

On the other hand, starting from x  x1 and proceeding in a similar way, we find for the Jost function ψ the following asymptotic behavior ψ(x, kj ) ∼ e

ikj x



0 IM

 j−1 Y

∗ ¯H a l (kj , Sl )

xj  x  xj−1

(4.130)

l=1 left

where the additional argument in ¯ a is to take into account the dependence on the norming constant (cf. (4.124)) and we have used (4.33) and the symmetry (4.66) relating c(k) to ¯ aH (k ∗ ). The notation left indicates that the product is performed such that the matrix with index ` occurrs to the left of the matrix with index ` − 1, i.e. j−1 Y

∗ ∗ ∗ ∗ ¯H a aH aH aH l (kj , Sl ) = ¯ j−1 (kj , Sj−1 )¯ j−2 (kj , Sj−2 ) . . . ¯ 1 (kj , S1 ).

l=1 left

If we restrict ourselves to the case N = 1, then al (kj ) are just scalars and therefore we write them simply as al (kj ); moreover eq. (4.59) yields φ(x, kj ) = ψ(x, kj )Cj a0 (kj ). Comparing (4.129) and (4.130), we get J Y

al (kj ) SH j =

j−1 Y

∗ 0 ¯ aH l (kj , Sl ) Cj a (kj )

(4.131)

l=1 left

l=j+1

and therefore 2

Cj (t) a0 (kj ) ∼ e4ikj t

J Y

al (kj )

l=j+1

j−1 Y

l=1 right


− −1 ∗ ¯ aH l (kj , Sl )

S− j


H

t → −∞

(4.132)

¯

4ikj Sj as t → −∞. Proceeding in a similar fashion as where S− j denotes the asymptotics of e t → +∞ and taking into account that the order of solitons is reversed, we get 2

Cj (t) a0 (kj ) ∼ e4ikj t

j−1 Y l=1

al (kj )

J Y

l=j+1 left


H
+ −1 ∗ ¯H a S+ l (kj , Sl ) j 77

t → +∞

(4.133)

and, taking into account (4.110), the comparison between the two representations (4.132)–(4.133) for Cj (t) yields J−1 Y

− S+ J = SJ

l=1

− S+ j = Sj

j−1 Y l=1

J−1 Y
−1 1 ¯ al (kJ∗ , S− l ) ∗ al (kJ )

(4.134)

l=1 left

j−1 J J Y Y Y
1 − −1 ∗ ∗ a (k ) ¯ al (kj∗ , S+ a ¯ (k , S ) l j l j l ) . l al (kj )∗ l=j+1

l=1 left

(4.135)

l=j+1 left

The relations (4.134) and (4.135) solve the problem of a J-soliton collision. Indeed, given S− l for + l = 1, . . . , J, (4.134) allows one to find S+ and since the expression (4.135) for S with j < J j J + + depends, through ¯ a, on Sl for l > j one can find iteratively Sj for any j. Let us consider the interaction of two solitons in more details. In this case from (4.134)–(4.135) it follows 1 − ¯ a (k ∗ , S− )−1 (4.136) S+ 2 = S2 a1 (k2 )∗ 1 2 1 − ∗ S+ a2 (k1∗ , S+ 1 = S1 a2 (k1 ) ¯ 2) .

(4.137)

Comparison of relations (4.134)–(4.135) with (4.136)–(4.137) indicates that a J soliton collision does not, in general, reduce to a pairwise collision. This is evident, for example, from the fact that + the expression for S+ ` contains Sj with j > `, which depend on the initial parameters of all the remaining solitons. Note that the formulae (4.136)–(4.137) are not symmetric with respect to the exchange of the subscripts 1 and 2. However, such a notation expresses the invariance of the system under the substitution t → −t, q → q∗ (here a “fast” soliton becomes a “slow” soliton and vice-versa). Taking into account the explicit expressions of a and ¯ a for the pure soliton case, one can solve (4.136) for S+ and then substitute it into the right-hand side of (4.137) in order to get S+ 2 1 . Indeed, (4.118) and (4.128) yield

and from (4.122)–(4.123) it follows aj (k) =

k − kj , k − kj∗

S±

j p± j = ± Sj

j = 1, 2

¯ aj (k, S± j ) = IM +

(4.138)

kj − kj∗ 1 ±

S k − kj∗ S± 2 j

H

S± j .

(4.139)

j

Inserting (4.139) into (4.136)–(4.137), the unit polarization vectors after the interaction in (4.138) can be expressed in terms of the parameters characterizing the colliding solitons as follows   1 k1 − k2∗ k1∗ − k1 − ∗ −
− + − p2 = p2 + ∗ (4.140) p · p2 p1 χ k1∗ − k2∗ k2 − k1∗ 1   k2∗ − k2 − ∗ −
− 1 k1 − k2∗ − p · p1 p2 p + (4.141) p+ = 1 1 χ k1 − k2 k2 − k1 2 where

( + S k1 − k ∗ (k1∗ − k1 ) (k2 − k2∗ ) − 2 2 χ = − = 1 + p1 2 k1 − k2 S2 |k1 − k2 | 78

∗

·

2 p− 2

)1/2

.

(4.142)

It is clear that the magnitude of the soliton polarizations do not change when their initial polarizations are either parallel or orthogonal. Indeed, without loss of generalization, one can parametrize the polarization vectors of the solitons as   p iϕ (4.143) p− p− 1 − ρ2 1 = (1, 0) , 2 = ρe , with 0 ≤ ρ ≤ 1 and 0 ≤ ϕ ≤ 2π, then from (4.140)–(4.141) the polarization vectors after the interaction are such that, for instance, 2 2 1 k1 − k2∗ k2∗ − k2 2 + (1) 2 ρ 1 + p1 = 2 χ k1 − k2 k2 − k1 4 1 k1 − k2∗ 2 + (1) 2 ρ p2 = 2 χ k1 − k2 )1/2 ( ∗ ∗ k1 − k2∗ (k − k ) (k − k ) 2 1 1 2 2 1+ χ = ρ 2 k1 − k2 |k1 − k2 |

+ (j)

where, as usual, p1 for j = 1, 2 denote the j-th component of the corresponding vector. − (1) 2 + (1) 2 In the polarization basis (4.143), we always have |p1 | = 1, thus the quantity |p1 | measures the shift in the intensity of soliton 1 that results from the interaction of the solitons. + (1) 2 + (1) 2 Specifically, 0 ≤ |p1 | ≤ 1, where |p1 | = 1 indicates no change in the intensity distribution + (1) 2 of soliton 1 and |p1 | = 0 indicates that all the intensity of soliton 1 is shifted to an orthogonal polarization by the soliton interaction. + (1) 2 In Fig. 4.1 we plot |pj | for j = 1, 2 as functions of ρ2 for some specific choice of the eigenvalues. Èp+1H1LÈ2

Èp+2H1LÈ2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.2

0.4

0.6

0.8

1

Ρ2

0.2

0.4

0.6

0.8

1

Ρ2

Figure 4.1: Intensity shift induced by two-soliton interaction of 2-component VNLS. The soliton parameters are: kj = −bj + iaj with a1 = 1, a2 = 2, b1 = ∆b, b2 = −∆b and ∆b = 1, .5, .1 from the top to the bottom for the plot to the left and viceversa for the one to the right. The initial polarizations are given by (4.143).

4.3

Conserved quantities and Hamiltonian structure

We showed that the scattering coefficient a(k) is time-independent. Since a(k) − IN is analytic in the upper half k-plane and a(k) − IN → 0 as |k| → ∞, it admits a Laurent expansion whose 79

coefficients are constants of the motion, as well. From the integral representation (4.43) for a(k), it follows that the quantities Z +∞ Γj = q(x)M(dn),−j (x)dx (4.144) −∞

are conserved for any integer j ≥ 0 and the coefficients M(dn),−j (x) of the asymptotic expansion of M(dn) (x, k) can be calculated iteratively from (4.16). For instance, M(dn),−1 (x) and is given by (4.22) and inserting the asymptotics (4.22) into (4.16), one finds Z x q(ξ)r(ξ)dξ − rx (x) M(dn),−2 (x) = r(x) −∞

etc. Therefore the first constants of the motion are given by Z +∞ q(x)r(x)dx Γ1 = − −∞ "Z # Z +∞

Γ2 =

ξ1

q(ξ2 )r(ξ2 )dξ2 −

dξ1 q(ξ1 )r(ξ1 )

−∞

Γ3 =

+∞

Z

(

ξ1

Z

q(ξ2 )r(ξ2 )

−∞

"Z

Z

+∞

q(x)rx (x)dx

(4.146)

#

(4.147)

−∞

−∞

q(ξ1 ) −r(ξ1 )

−∞

(4.145)

ξ2

q(ξ3 )r(ξ3 )dξ3 dξ2

−∞

" # ) Z ξ1 ∂ + r(ξ1 ) q(ξ2 )r(ξ2 )dξ2 − rξ1 ξ1 (ξ) dξ1 ∂ξ1 −∞ and so on. The scattering coefficient ¯ a(k) is also a constant of the motion and proceeding exactly as before one can obtain a second set of conserved quantities given by Z +∞ ¯ ¯ (up),−j (x)dx r(x)M Γj = −∞

for any j ≥ 1, yielding Z ¯ Γ1 =

+∞

r(x)q(x)dx

(4.148)

−∞

¯2 = Γ

Z

+∞

−∞

dξ1 r(ξ1 )q(ξ1 )

"Z

#

ξ1

r(ξ2 )q(ξ2 )dξ2 −

−∞

Z

+∞

r(x)qx (x)dx

(4.149)

−∞

and so on. With the reductions r = ∓qH for N = 1, q(x)r(x) is a scalar and the constants of the motion in (4.145)–(4.148) can be simplified as follows Z Z Γ1 = ± kq(x)k dx, Γ2 = ± q(x)qH (4.150) x (x)dx Z   Γ3 = ± kqx (x)k + kq(x)k4 dx. (4.151) 80

The dynamical system (4.5) is Hamiltonian, as may be seen from the identification: coordinates (q) : momenta (p) : Hamiltonian (H) :

q(j) (x, t) q(j) (x, t)∗ Z +∞   2 4 ±i kqx k + kqk dx −∞

with the canonical brackets o n q(j) (x, t), q(k) (y, t)∗ = iδ(x − y)δj,k n o q(j) (x, t), q(k) (y, t) = 0.

Note that the Hamiltonian is given by the conserved quantity (4.151).

81

Chapter 5

Discrete matrix NLS 5.1

Overview

The integrable discrete matrix NLS system d Qn = Qn+1 − 2Qn + AQn + Qn B+Qn−1 − Qn+1 Rn Qn − Qn Rn Qn−1 dτ d −i Rn = Rn+1 − 2Rn + BRn + Rn A+Rn−1 − Rn+1 Qn Rn − Rn Qn Rn−1 dτ i

(5.1) (5.2)

where Qn and Rn are, respectively, N × M and M × N matrices, results to from the compatibility condition of the following linear equations   zIN Qn vn (5.3) vn+1 = Rn z −1 IM d vn = dτ

!
2 −izQn + iz −1 Qn−1 iQn Rn−1 − 2i z − z −1 IN − iA
2 vn iz −1 Rn − izRn−1 −iRn Qn−1 + 2i z − z −1 IM + iB

(5.4)

where, as usual, IN is the N × N identity matrix and IM is the M × M identity matrix. Note that the matrices A and B in (5.4) and in the system (5.1)–(5.2) can be absorbed by the gauge transformation  iτ A  e 0 iτ A iτ B −iτ B −iτ A ˆ ˆ Qn = e Qn e , R n = e Rn e , v ˆn = vn , (5.5) 0 e−iτ B that is, (5.3)–(5.4) are satisfied under the substitutions ˆ n, Qn → Q

ˆ n, Rn → R

vn → v ˆn ,

A, B → 0.

The system (5.1)–(5.2) does not, in general, admit the reduction Rn = ∓QH n , and the asymmetry depends on the non-commutativity of matrix multiplication. However, if one takes M = N and restricts Rn and Qn to be such that Rn Qn = Qn Rn = αn IN

82

(5.6)

and αn is real when Rn = ∓QH n , then this symmetry is a consistent reduction of (5.1)–(5.2), which reduces to the single (matrix) equation i

d Qn = Qn+1 − 2Qn + AQn + Qn B + Qn−1 − αn (Qn+1 + Qn−1 ) . dτ

(5.7)

In the case N = 2, the matrices (1)

Qn =

!

(2)

Qn Qn (2) (1) (−1)n Rn (−1)n+1 Rn

(1)

,

Rn =

(2)

Rn (−1)n Qn (2) (1) Rn (−1)n+1 Qn

!

(5.8)

satisfy the condition (5.6) with   (2) (2) I. + R Q Rn Qn = Qn Rn = αn I = Rn(1) Q(1) n n n

Note that (5.8) is equivalent to

Rn = (−1)n PQTn P

(5.9)

where P=



0 1 −1 0



(5.10)

and the superscript T denotes matrix transposition. We also note that the symmetry (5.6) between Qn and Rn imposes restrictions on the choice of the gauge matrices A and B. Indeed, if the symmetry condition (5.9) is imposed at the initial time, say τi = 0, Rn (0) = (−1)n PQTn (0)P

(5.11)

it is preserved by the time evolution if, and only if,  d  n T Rn (τ ) − (−1) PQn (τ )P =0 dτ τ =0

(5.12)

that is, taking into account (5.1)–(5.2), for A and B satisfying B = 2I + PBT P,

A = 2I + PAT P.

(5.13)

The condition (5.13) is accomplished, for instance, if A = B = I. In this case, in order to obtain ˆ n and R ˆn solutions of the system (5.1)–(5.2) one first determines the evolution of the potentials Q such that ˆ n (0) = Qn (0) , Q

ˆ n (0) = Rn (0) R

which solve the system (5.1)–(5.2) with gauge A = B = I, i.e. d ˆ ˆ n+1 + Q ˆ n−1 − Q ˆ n+1 R ˆ nQ ˆn − Q ˆ nR ˆ nQ ˆ n−1 Qn = Q dτ d ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ −i R n = Rn+1 + Rn−1 − Rn+1 Qn Rn − Rn Qn Rn−1 dτ i

83

(5.14) (5.15)

and then uses the transformations ˆ n (τ ) eiτ B = e2iτ Q ˆ n (τ ) Qn (τ ) = eiτ A Q ˆ n (τ ) e−iτ A = e−2iτ R ˆ n (τ ) Rn (τ ) = e−iτ B R to obtain the solution of (5.1)–(5.2) with gauge A = B = 0. Note that the gauges     00 00 A= , B= , 04 00

(5.16) (5.17)

(5.18)

also satisfy the condition (5.13) and with the rescaling (j) Q(j) n = hqn ,

Rn(j) = hrn(j) ,

τ = h−2 t

(5.19)

the system (5.1)–(5.2) with A and B given by (5.18) reduces to the two-component symmetric   T T (1) (2) (1) (2) . and rn = rn , rn system (1.9)–(1.10) where qn = qn , qn The symmetry (5.6) that has no counterpart in the IST for the scalar IDNLS. The symmetry between the matrices induces the additional symmetry in the scattering data associated with the block-matrix scattering problem (see the Sec. on Symmetries below). This additional symmetry must be included in the formulation of the block-matrix inverse scattering problem in order to obtain a solution of the system (5.1)–(5.2) via the IST.

5.2 5.2.1

The Inverse Scattering Transform Direct Scattering Problem

Jost functions The scattering problem (5.3) associated with integrable discrete matrix NLS can be written compactly as vn+1 = Sn vn

(5.20)

where Sn =



zIN Qn Rn z −1 IM



.

(5.21)

The solutions of this difference equation have the asymptotic boundary conditions     IN ¯n (z) ∼ z −n 0 φn (z) ∼ z n , φ as n → −∞ IM 0

(5.22)

and ψ n (z) ∼ z

−n



0 IM



,

¯n (z) ∼ z n ψ



IN 0



as n → +∞

(5.23)

which are determined by the boundary condition as n → ±∞ respectively. These solutions are matrix-valued functions with the following dimensions: ¯n (z) : (N + M ) × M φ ¯n (z) : (N + M ) × N. ψ

φn (z) : (N + M ) × N, ψ n (z) : (N + M ) × M, 84

¯ and ψ, ψ ¯ form We will prove in the following (see the section on Scattering Data) that both φ, φ a basis of solutions As in the IST for the 2 × 2 scattering problem we dealt with in Chap. 3, we consider the Jost functions which have constant (i.e. independent of n and z) boundary conditions as n → ±∞. The Jost functions are related to the eigenfunctions via the following transformations: ¯n (z), ¯ n (z) = z n φ M −n ¯ n (z) = z ψ ¯n (z). N

Mn (z) = z −n φn (z), n

Nn (z) = z ψ n (z),

Hence, the Jost functions satisfy the constant boundary conditions:     IN 0 ¯ n (z) ∼ Mn (z) ∼ , M as n → −∞ 0 IM

(5.24) (5.25)

(5.26)

and Nn (z) ∼



0 IM



,

¯ n (z) ∼ N



IN 0



as n → +∞

(5.27)

and the difference equations: ˜ n Mn (z) Mn+1 (z) − z −1 AMn (z) = z −1 Q ¯ n+1 (z) − zAM ¯ n (z) = z Q ˜ nM ¯ n (z) M ˜ n Nn (z) Nn+1 (z) − zANn (z) = z Q −1 −1 ˜ ¯ ¯ ¯ Nn+1 (z) − z ANn (z) = z Qn Nn (z)

(5.28) (5.29) (5.30) (5.31)

where A=



zIN 0 0 z −1 IM



˜n = Q

,



0 Qn Rn 0



.

The difference equations (5.28)–(5.31), along with their respective boundary conditions (5.26)– (5.27), can be converted to summation equations by the method of Green’s functions. The derivation of the Green’s functions is essentially the same as for the 2×2 scattering problem. The Green’s function corresponding to (5.28) or, equivalently, (5.31), is a solution of the difference equation Gn+1 − z −1 AGn = z −1 δ0,n IN .

(5.32)

If vn satisfies the summation equation vn = w +

+∞ X

˜ k vk Gn−k Q

(5.33)

k=−∞

where Gn is a solution of (5.32) and w satisfies w − z −1 Aw = 0,

(5.34)

then vn is a solution of the difference equation (5.28) or, equivalently, (5.31). The Green’s function is not unique, and, as we show below, the choice of the Green’s function and the choice of the inhomogeneous term w together determine the Jost function and its analytical properties. To find the Green’s function explicitly, we first note that one can set the off-diagonal blocks of Gn to zero and write ! (1) gn IN 0 (5.35) Gn = (2) 0 gn IM 85

(j)

where, according to (5.32), gn must satisfy (j)

gn+1 − b(j) gn(j) = z −1 δ0,n

j = 1, 2

(5.36)

with b(1) = 1,

b(2) = z −2 .

(5.37)

(j)

Next, we represent gn and δ0,n as Fourier integrals I 1 (j) gn = pn−1 gˆ(j) (p)dp, 2πi |p|=1

δ0,n

1 = 2πi

I

pn−1 dp.

|p|=1

Substituting these integrals into the difference equations (5.36) yields gˆ(j) (p) = z −1

1 p − b(j)

and gn(j)

=z

−1

1 2πi

I

|p|=1

1 pn−1 dp. p − b(j)

(5.38)

(j) The integral in (5.38) depends only whether the pole (1) b (2)is located inside or outside the contour of integration. However, when |z| = 1 we have b = b = 1, i.e. the poles are on the contour |p| = 1. As usual, we consider contours that are perturbed away from |p| = 1 in order to avoid the singularities. Let C out be a contour enclosing p = 0 and p = b(j) and let C in be a contour enclosing p = 0 but not p = b(j) (see Fig. 3.1). Consequently, we get

gn(j),out

=z

−1

1 2πi

I

C out

1 pn−1 dp = z −1 p − b(j)



b(j) 0


n−1

n≥1 n≤0

(5.39)

and gn(j),in = z −1

1 2πi

I

C in

1 pn−1 dp = z −1 p − b(j)



0
n−1 − b(j)

n≥1 . n≤0

(5.40)

By substituting one or the other of (5.39) or (5.40) into (5.35), with b(j) given by (5.37), we obtain two Green’s functions satisfying (5.36), i.e.   IN 0 ` −1 (5.41) Gn (z) = z θ(n − 1) 0 z −2(n−1) IM ¯ rn (z) = −z −1 θ(−n) G



IN 0

0 z −2(n−1) IM



where θ(n) is the discrete version of the step function introduced in (3.32), i.e. θ(n) =

n X

δ0,k =

k=−∞

86



1 0

n≥0 . n<0

(5.42)

Taking into account the boundary conditions (5.26)–(5.27) and the relation (5.34) for the inhomogeneous term in (5.33), if Qn , Rn → 0 as n → ±∞ we get the following summation equations ¯n for Mn and N Mn = ¯n = N



IN 0



+



IN 0



+

+∞ X

˜ k Mk G`n−k Q

(5.43)

¯ rn−k Q ˜ kN ¯k G

(5.44)

k=−∞ +∞ X

k=−∞

¯ n (z) and Nn (z) A similar approach yields the summation equations for M ¯n = M Nn =



0 IM



+



0 IM



+

+∞ X

¯ `n−k Q ˜ kM ¯k G

(5.45)

˜ k Nk . Grn−k Q

(5.46)

k=−∞ +∞ X

k=−∞

where Grn (z) = −θ(−n)z −1 ¯ rn (z) = −θ(−n)z −1 G

 

z 2(n−1) IN 0 0 IM



IN 0 0 z −2(n−1) IM



(5.47) .

(5.48)

Existence and Analyticity of the Jost functions The existence of the Jost functions P can be proved by iteration (i.e. Neumann series) as long as kQk1 , kRk1 < ∞. Here kQk1 = +∞ n=−∞ kQn ka , where kQn ka is any matrix norm. We also note that, if the potentials Qn and Rn have finite support (Qn ,Rn = 0 if n < nmin and n > nmax ), each of the Jost functions can be generated by a finite number of applications of the appropriate difference equation (5.28)–(5.31) with the proper choice of boundary condition. Hence, in the following we assume that the potentials are such that Jost functions exist. In order to prove the existence and the sectional analyticity of the Jost functions we will make use of Lemma A1 and Lemma A2 proved in Appendix A. First we consider the Jost function Mn (z). The Neumann series Mn (z) =

∞ X

Cjn (z)

(5.49)

j=0

where C0n (z) = Cj+1 n (z) =



IN 0

+∞ X

 ˜ k Cj (z), G`n−k (z)Q k

j≥0

(5.50)

k=−∞

is, formally, a solution of the summation equation (5.43). To make this rigorous, we establish a bound on the Cjn such that the series representation (5.49) converges absolutely and uniformly in n and absolutely and uniformly in z in the region |z| ≥ 1. 87

The summation equation for Mn (z) can be written as n−1 X

M(up) (z) = IN + z −1 n

(dn)

Qk Mk

(z)

(5.51)

k=−∞ n−1 X

M(dn) (z) = n

(up)

z −2(n−k−1)−1 Rk Mk

(z)

(5.52)

k=−∞

and eq. (5.50) in upper/lower component form is Cnj+1,(up) (z) = z −1

n−1 X

j,(dn)

Q k Ck

n−1 X

Cnj+1,(dn) (z) = z −1

(z),

j,(up)

z −2(n−1−k) Rk Ck

(z).

k=−∞

k=−∞

0,(dn)

2j+1,(up)

2j,(dn)

Then, since Cn = 0, it follows that also Cn 2j,(up) 2j+1,(dn) only need to find bounds for Cn and Cn . We prove by induction that for |z| ≥ 1 P n−1

k=−∞ kRk ka



2j+1,(dn) (z) ≤

Cn

(j + 1)!

a

P n−1

k=−∞



2(j+1),(up) (z) ≤

Cn

kRk ka

(j + 1)!

a

(5.53)

= 0, Cn

j+1 P n−1

= 0 for any j ≥ 0 and we

k=−∞ kQk ka

j+1 P n−1

j!

k=−∞

kQk ka

(j + 1)!

j

(5.54)

j+1

.

(5.55)

The proof follows from the inductive step through the application of (5.53) twice. One iteration yields n−1

X

2j+1,(dn) (z) ≤ kRk ka

Cn a

k=−∞

≤

P n−1

P k−1

`=−∞

k=−∞ kQk ka

j!

j

kR` ka

j! n−1 X

j P k−1

`=−∞

P k−1

kQ` ka

j!

j

`=−∞ kR` ka

kRk ka

k=−∞

j!

j

then, applying Lemma A.2 completes the induction for (5.54). The next iteration of (5.53) yields for |z| ≥ 1 n−1



X

2j+1,(dn)

2j+2,(up) −1 (z) kQk ka Ck (z) ≤ |z|

Cn a

a

k=−∞

≤

n−1 X

kQk ka

k=−∞

≤

P n−1

k=−∞

P k−1

`=−∞ kQ` ka

kRk ka

(j + 1)!

j+1

j! n−1 X

k=−∞

and we again use Lemma A2 to complete the induction. 88

j P k−1

`=−∞ kR` ka

(j + 1)!

kQk ka

P k−1

`=−∞

j+1

kQ` ka

j!

j

The bounds (5.54)–(5.55) are absolutely and uniformly (in n) summable if kQk1 , kRk1 < ∞ where k·k1 is the L1 norm. Moreover, in this case, the Neumann series (5.49) converges uniformly for all |z| ≥ 1. Hence, the Jost function Mn (z) exists and is continuous in the region |z| ≥ 1. Note that we originally constructed Mn (z) for |z| = 1 but the summation equation allows us to extend Mn (z) to the region |z| ≥ 1. Because the Neumann series (5.49) converges absolutely, this yields a convergent Laurent series in powers of z −1 for the solution Mn (z). Thus, the bounds (5.54)–(5.55) estabilish that Mn (z) is analytic in the region |z| > 1 when Qn , Rn ∈ L1 . Analogously, one can prove the existence and analyticity of the remaining Jost functions. ¯ n (z), the maps Note that in order to generate Nn (z) and N Nn (z) = z −1 S−1 n (z)Nn+1 (z) −1 ¯ ¯ n+1 (z) Nn (z) = zSn (z)N where Sn (z) is given by (5.21), must be well-defined because these Jost functions are determined by a boundary condition in the limit n → +∞. That is, the matrix Sn (z) must be invertible. Equivalently, it must be that det Sn = z N −M det (IM − Rn Qn ) = z N −M det (IN − Qn Rn ) 6= 0

(5.56)

where we used the Schur identities to compute the deteminant of the block-partioned matrix Sn (see [36]). Specifically, for a matrix ∆ partitioned in four blocks   AB ∆= (5.57) CD where A and D are square matrices, we have det ∆ = det A det D − CA−1 B −1

det ∆ = det D det A − BD




C




if

det A 6= 0

(5.58)

if

det D 6= 0.

(5.59)

Note that (5.56) is reminiscent of the condition 1 − Rn Qn 6= 0 that was required in the 2 × 2 scattering problem. This condition holds if, for example, Qn and Rn are sufficiently small (in any matrix norm) or if all the eigenvalues of the product Rn Qn are negative. From here on, we assume that Qn , Rn satisfy (5.56). Properties of the Jost functions in the Complex z-Plane ¯ n (z) When the potentials have finite support, Mn (z) and Nn (z) are polynomials in z −1 while M ¯ n (z) are polynomials in z for each finite n. Therefore, all of the Jost functions are analytic and N everywhere in z, except possibly at z = 0 and z = ∞. Moreover, we showed that the Jost functions are analytic in the region |z| > 1 or in the region |z| < 1 if the potentials have a finite L1 norm. If we assume that Nn (z) and Mn (z) are analytic in the region |z| > K, then they have Laurent ¯ ¯ series expansions in z −1 which converge in the region of analyticity. Similarly, N(z) and M(z) are ¯ and then they have power series expansions in z which converge in analytic in the region |z| < K the region of analyticity. In particular, when the potentials have finite support, these expansions terminate and the Jost functions are polynomials in z −1 or in z. Let us write the Laurent series expansion for Mn (z) as M(up) (z) = n

+∞ X

z −j Mn(up),−j

(5.60)

z −j Mn(dn),−j

(5.61)

j=0

M(dn) (z) = n

+∞ X j=0

89

and the summation equation (5.43) for Mn (z) in the form n−1 X

M(up) (z) = IN + n

(dn)

z −1 Qk Mk

(z)

(5.62)

k=−∞ n−1 X

M(dn) (z) = n

(up)

z −2(n−k)+1 Rk Mk

(z).

(5.63)

k=−∞

If we substitute the expansions into (5.62)–(5.63) and match the terms of O(1), we get M(up),0 = IN , n

M(dn),0 = 0, n

which anchors an induction on the coefficients of the expansion. Indeed, one can show by induction that for any integer j ≥ 0 Mn(up),−2j = 0.

M(up),−(2j+1) = 0, n

Moreover, matching the corresponding powers of z −1 into the summation equations (5.62)–(5.63) yields the following recursive relations M(dn),−2j+1 = n

n−1 X

(up),−2(j+k−n)

(5.64)

(dn),−2j+1

(5.65)

Rk Mk

k=n−j

Mn(up),−2j

=

n−1 X

Qk Mk

.

k=−∞

¯ n (z) about z = 0, i.e. Analogously, for the coefficients of the power series expansion of M ¯ (up) M (z) = n

+∞ X

¯ (up),j zj M , n

¯ (dn) M (z) = n

j=0

n−1 X

¯ (dn) zj M n

(5.66)

k=−∞

one obtains the following ¯ (dn),0 M = IM . n

¯ (up),0 M = 0, n

Then it is easy to prove by induction that for any integer j ≥ 1 ¯ (dn),2j−1 M =0 n

¯ n(up),2j = 0, M and ¯ (up),2j−1 M = n

n−1 X

¯ (dn),2(j−n+k) Qk M k

(5.67)

¯ (up),2j−1 . Rk M k

(5.68)

k=n−j

¯ n(dn),2j = M

n−1 X

k=−∞

Therefore we can write IN + O(z −2 , even) Mn (z) = z −1 Rn−1 + O(z −3 , odd)   zQn−1 + O(z 3 , odd) ¯ Mn (z) = , IM + O(z 2 , even) 

90



(5.69) (5.70)

where “even” indicates that the remaining terms are even powers of z and “odd” indicates that the remaining terms are odd powers. ¯ n (z) are obtained from the summation equations (5.46) The expansions in z of Nn (z) and N and (5.44) respectively. We write the summation equation (5.46) as N(up) (z) = − n

∞ X

(dn)

z −2(k−n)−1 Qk Nk

(z)

(5.71)

k=n

N(dn) (z) n

= IM −

∞ X

(up)

zRk Nk

(z)

(5.72)

k=n

and we substitute the expansions N(up) (z) = n

+∞ X

z −j Nn(up),−j

j=0

N(dn) (z) = n

+∞ X

z −j Nn(dn),−j

j=0

into (5.71)–(5.72) to obtain the equations N(up),0 =0 n

(5.73)

Nn(up),−1 = −Qn N(dn),0 n Nn(up),−2 = −Qn Nn(dn),−1

N(dn),0 = IM − n Nn(dn),−1 = −

+∞ X

(up),−1

Rk Nk

k=n +∞ X

(up),−2

Rk Nk

.

(5.74) (5.75)

k=n

Further manipulation yields (dn),0

Nn+1 = (IM − Rn Qn ) N(dn),0 n (dn),−1

Nn+1

= (IM − Rn Qn ) Nn(dn),−1 .

Solving these equations with the boundary condition (5.27) yields −1

N(dn),0 = (IM − Rn Qn ) (IM − Rn+1 Qn+1 ) n +∞ Y −1 (IM − Rk Qk ) =

−1

...

k=n right

where the term “right” in the product indicates that the matrix with index k occurs to the right of the matrix with index k − 1. Also, Nn(dn),−1 = 0. By substituting these expressions back into (5.74)–(5.75) we obtain   −1 −3 −z Qn ∆−1 ) n + O(z Nn (z) = −2 ) ∆−1 n + O(z 91

(5.76)

where ∆n = . . . (IM − Rn+1 Qn+1 ) (IM − Rn Qn ) +∞ Y (IM − Rk Qk ) =

(5.77)

k=n left

and “left” indicates that the matrix with index k occurs to the left of the matrix with index k − 1. ¯ n (z) in powers of z Analogously, we derive the expansion for N   2 Ω−1 n + O(z ) ¯ Nn (z) = (5.78) 3 −zRn Ω−1 n + O(z ) where Ωn =

+∞ Y

(IN − Qk Rk )

(5.79)

k=n left

and, as before, “left” indicates that the matrix with index k occurs to the left of the matrix with index k − 1. Scattering data ¯n (z) and ψ n (z) together constitute N + M solutions of the difference equation The Jost functions ψ (5.20). To show that these solutions are linearly independent for all n, we calculate their Wronskian. The matrix
¯n (z , ψ n (z)) Ψn (z) = ψ

is an (N + M ) × (N + M ) square matrix. Hence, we define
¯n (z), ψ n (z) = det Ψn (z) W ψ

(5.80)

and similarly for any collection of matrices which, all together, have the same number of rows and columns. In particular,

¯n+1 (z), ψ n+1 (z) = W Sn (z)ψ ¯n (z), Sn (z)ψ n (z) W ψ
¯n (z), ψn (z) = det Sn (z)W ψ
¯n (z), ψn (z) = z N −M det (IM − Rn Qn ) W ψ and therefore for any j ∈ N0


¯n (z), ψ n (z) = Q W ψ j−1 l=0

= Qj−1 l=0

z j(M−N ) det (IM − Rn+l Qn+l ) z n(N −M) det (IM − Rn+l Qn+l )

Taking the limit as j → ∞, we obtain


¯n+j (z), ψn+j (z) W ψ


¯ n+j (z), Nn+j (z) . W N

z n(N −M) j=n det (IM − Rj Qj )


¯n (z), ψ n (z) = Q∞ W ψ 92

(5.81)

which is nonzero if det (IM − Rn Qn ) 6= 0, a condition that we have already assumed. ¯n (z) constitute a second set of M + N solutions of the scattering The functions φn (z) and φ problem (5.20) and n−1 Y
¯n (z) = z n(N −M) W φn (z), φ det (IM − Rj Qj ) .

(5.82)

j=−∞

¯n (z) are linearly dependent on the set of solutions ψ n (z) and Hence, the solutions in φn (z) and φ ¯ ψ n (z). This dependence can be expressed as ¯n (z)a(z) φn (z) = ψ n (z)b(z) + ψ ¯n (z) = ψ n (z)¯ ¯n (z)b(z) ¯ φ a(z) + ψ

(5.83) (5.84)

¯ where b(z) is an M × N matrix, a(z) is an N × N matrix, ¯ a(z) is an M × M matrix and b(z) is an N × M matrix. In block-matrix form, (5.83)–(5.84) are   ¯

ab ¯ ¯ φn , φn = ψ n , ψ n (5.85) b¯ a and in terms of the Jost functions, eqs. (5.83)–(5.84) are

¯ n (z)a(z) Mn (z) = z −2n Nn (z)b(z) + N ¯ ¯ n (z) = Nn (z)¯ ¯ n (z)b(z). M a(z) + z 2n N

(5.86) (5.87)

¯ These equations define the coefficients a(z), ¯ a(z), b(z) and b(z) for any z such that all four Jost functions exist. The scattering equations (5.83)–(5.84) have the counterpart ¯n (z)c(z) ψ n (z) = φn (z)d(z) + φ ¯ ¯ ¯ ψ n (z) = φn (z)¯ c(z) + φn (z)d(z)

(5.88) (5.89)

which express the eigenfunctions with boundary conditions defined as n → +∞ (i.e. ψ n (z) and ¯n (z)) in terms of eigenfunctions with boundary conditions defined as n → −∞ (i.e. φn (z), φ ¯n (z)). ψ In block-matrix form  
¯
cd ¯ ¯ (5.90) ψ n , ψ n = φn , φn ¯c d and by comparing (5.85) and (5.90) we get the relation between “left” and “right” scattering data 

¯ ab b¯ a

−1

=



¯ cd ¯c d



(5.91)

which is valid for any z such that all the scattering coefficients are well-defined. In particular it holds for |z| = 1, i.e. for all z such that z = 1/z ∗. The scattering coefficients a(z) and b(z) can be written as explicit sums of the Jost function ˜ n . The formula is derived as follows. First, we obtain the relation Mn (z) and the potential Q ¯ n (z)a(z) = Mn (z) − N 

IN − a(z) 0



+

+∞ n o X ˜ k Mk (z) − G ¯ rn−k (z)Q ˜ kN ¯ k (z)a(z) G`n−k (z)Q

k=−∞

93

by substituting the right-hand sides of the summation equations (5.43) and (5.44) for Mn (z) and ¯ n (z) respectively. Then, we use the identity N   IN 0 ` r −1 ¯ Gn (z) = Gn (z) + z 0 z −2(n−1) IM ¯ n (z)a(z) with z −2n Nn (z)b(z), so that and the relation (5.86) to replace Mn (z) − N z −2n Nn (z)b(z) = 

IN − a(z) 0



+

+∞ X

¯ rn−k (z)Q ˜ k Nk (z)b(z) z −2k G

k=−∞

 +∞  −1 X z IN 0 ˜ k Mk (z). Q + 0 z −2(n−k)+1 IM k=−∞

¯ rn (z) = z −2n Grn (z), we obtain Finally, taking into account that G ( ) +∞ X r −2n ˜ k Nk (z) b(z) = Nn (z) − Gn−k (z)Q z k=−∞



IN − a(z) 0



+

 +∞  −1 X z IN 0 ˜ k Mk (z). Q 0 z −2(n−k)+1 IM

k=−∞

Assuming that the summation equation (5.46) for Nn (z) has unique solution, the term in curly braces is (0, IM )T and then we have 

a(z) z −2n b(z)



=



IN 0



 +∞  −1 X z IN 0 ˜ k Mk (z) Q + 0 z −2(n−k)+1 IM k=−∞

so that finally we obtain a(z) = IN +

+∞ X

(dn)

z −1 Qk Mk

(z)

(5.92)

k=−∞

b(z) =

+∞ X

(up)

z 2k+1 Rk Mk

(z).

(5.93)

k=−∞

¯ ¯ The same approach works for ¯ a(z) and b(z) and for c(z), d(z), ¯ c(z) and d(z). The corresponding

94

expressions are: +∞ X

¯ a(z) = IM +

(up)

¯ zRk M k

(z)

(5.94)

¯ (dn) (z) z −2k−1 Qk M k

(5.95)

k=−∞

¯ b(z) =

+∞ X

k=−∞ +∞ X

c(z) = IM −

(up)

zRk Nk

(z)

(5.96)

k=−∞

d(z) = −

+∞ X

(dn)

z −2k−1 Qk Nk

(z)

(5.97)

¯ (dn) (z) z −1 Qk N k

(5.98)

k=−∞

¯ c(z) = IN −

+∞ X

k=−∞

¯ d(z) =−

+∞ X

¯ (up) (z). z 2k+1 Rk N k

(5.99)

k=−∞

The expressions (5.92) and (5.94) imply that a(z) is analytic in the same region as Mn (z), i.e. ¯ n (z), i.e. |z| < 1. Similarly, (5.96) and (5.98), |z| > 1, and ¯ a(z) is analytic in the same region as M together with (5.76) and (5.78), imply that c(z) is analytic in the same region as Nn (z) (and, ¯ n (z). Inserting the expansions consequently, as a(z)) and ¯ c(z) is analytic in the same region as N (5.69)–(5.70), (5.76) and (5.78) for the Jost functions into the summation representations (5.92)– (5.98), we obtain the expansions for the scattering coefficients a(z), ¯ a(z), c(z) and ¯ c(z) a(z) = IN + O(z −2 , even),

¯ c(z) = IN +

+∞ X

2 Rk Qk ∆−1 k + O(z , even)

(5.100)

−2 Qk Rk Ω−1 , even) k + O(z

(5.101)

k=−∞

¯ a(z) = IM + O(z 2 , even),

c(z) = IM +

+∞ X

k=−∞

where ∆n and Ωn are given by (5.77) and (5.79). Hence a(z), ¯ a(z) and c(z), ¯ c(z) are even functions of the spectral parameter z. Moreover, the expressions (5.93), (5.95), (5.97) and (5.99) imply that ¯ ¯ b(z), b(z) and d(z), d(z) are odd functions of z. ¯ n (z), it is convenient to define Together with the Jost functions Mn (z) and M ¯ n (z)¯ µ ¯n (z) = M a−1 (z).

µn (z) = Mn (z)a−1 (z),

(5.102)

Note that µn (z) and µ ¯ n (z) are meromorphic in z and have poles where, respectively det a(z) = 0 and det ¯ a(z) = 0. In terms of these functions, the relations (5.86)–(5.87) are ¯ n (z) = z −2n Nn (z)ρ(z) µn (z) − N ¯ n (z)ρ(z) ¯ µ ¯n (z) − Nn (z) = z 2n N

(5.103) (5.104)

where the reflection coefficients ¯ a(z)−1 ρ(z) ¯ = b(z)¯

ρ(z) = b(z)a(z)−1 , are part of the n-independent scattering data. 95

(5.105)

The system (5.103)–(5.104) is the starting point for the inverse problem. Indeed, if ρ(z) and ρ(z) ¯ are known functions on the unit circle |z| = 1, then (5.103)–(5.104) is the boundary condition of a Riemann-Hilbert problem which we use to compute the functions µn (z), Nn (z) in the region ¯ n (z) in the region |z| < 1. |z| > 1, µ ¯ n (z) and N Proper eigenvalues Just as for the scalar case, we define a proper eigenvalue to be a value of z such that the scattering problem (5.20) has a solution that decays as n → ±∞. If |z| > 1, then the solutions φn (z) decay ¯ (z) as n → −∞ while the solutions ψ n (z) decay as n → +∞. On the other hand, the solutions φ n ¯ ¯ blow up as n → +∞ and the solutions ψ n (z) blow up as n → −∞. Recall that φn (z), φn (z) and ¯n (z) are both bases of solutions of the scattering problem. Therefore, if |zj | > 1 is an ψ n (z), ψ eigenvalue, it must be that one of the solutions in the span of φn (z) is in the span of ψ n (z). That is, zj is an eigenvalue if, and only if, W (φn (zj ), ψ n (zj )) = 0.

(5.106)

Similarly, z¯` , |¯ z` | < 1, is an eigenvalue if and only if
¯n (¯ ¯n (¯ W φ z` ), ψ z` ) = 0.

(5.107)

From the other side, the Wronskian (5.80) of the Jost solutions can be related to the scattering coefficient a(z) as follows:     
a(z)
0 ¯ ¯ ψ n (z), ψ n (z) , ψ n (z), ψn (z) W (φn (z), ψ n (z)) = W b(z) IM   ¯n (z), ψ n (z)) det a(z) 0 = W (ψ b(z) IM z n(N −M) det a(z) j=n det (IM − Rj Qj )

= Q+∞

(5.108)

showing that the eigenvalues in the region |z| > 1 are the points z = zj such that det a(zj ) = 0. Similarly, one can show that
z n(N −M) ¯n (z), φ ¯n (z) = Q W ψ det ¯ a(z) +∞ j=n det (IM − Rj Qj )

(5.109)

and, therefore, the eigenvalues in the region |z| < 1 are the points z = z¯` such that det ¯ a(¯ z` ) = 0. There are no eigenvalues on the circle |z| = 1 because, on this circle, none of the basis eigenfunctions vanish as n → ±∞. We also assume that det a(z), det ¯ a(z) 6= 0 for |z| = 1. Moreover, we assume that there is a finite number of eigenvalues in the regions |z| > 1 and |z| < 1. Previously, we showed that a(z) and ¯ a(z) are even functions of z. The determinants of these functions are therefore also even functions of z. In particular, if det a(zj ) = 0, then det a(−zj ) = 0 and the eigenvalues appear in pairs (the same holds for the eigenvalues |¯ zj | < 1). Without loss of generality, we will denote by zj the eigenvalues such that |arg zj | < π2 and −zj will be understood to be the eigenvalue such that |arg(−zj )| > π2 . If |arg zj | = π2 we consider zj to be the eigenvalue such that arg zj = π2 and therefore arg(−zj ) = − π2 . The same convention holds for the eigenvalues z¯` . The function µn (z) has poles precisely at the points z = zj such that det a(zj ) = 0. Similarly, the function µ ¯ n (z) has poles precisely at the points z = z¯` such that det a(¯ z` ) = 0. That is, the 96

eigenvalues of the scattering problem are the poles of the meromorphic functions µn (z) and µ ¯n (z) in the regions |z| > 1 and |z| < 1 respectively. We now obtain the expressions for the residues of these poles. If the pole at z = zj is simple (equivalently, the zero of det a(zj ) is simple), then Res (µn ; zj ) = lim (z − zj ) µn (z). z→zj

The relation (5.103) holds for any z such that all the Jost functions exist, therefore, if we write a−1 (z) =

1 α(z) a(z)

where a(z) = det a(z) and α(z) is the cofactor matrix of a(z), then  ¯ n (z) + (z − zj )z −2n Nn (z)ρ(z) Res (µn ; zj ) = lim (z − zj ) N z→zj

= zj−2n Nn (zj ) lim (z − zj )ρ(z) z→zj

=

zj−2n Nn (zj )Cj .

(5.110)

The M × N matrix Cj = lim (z − zj )ρ(z) = z→zj

1 b(zj )α(zj ) a0 (zj )

(5.111)

is the norming constant that relates Res(µn ; zj ) and Nn (zj ). By a similar procedure, we can find an expression for the residues of the poles of µ ¯n (z) in the region |z| < 1. We write ¯ a(z) =

1 α(z) ¯ a ¯(z)

where a ¯(z) = det ¯ a(z) and α(z) ¯ is the cofactor matrix of ¯ a(z). If we assume z¯` to be a simple zero of a ¯(z), then ¯ n (¯ ¯` Res (µ ¯n ; z¯` ) = z¯`2n N z` )C

(5.112)

where, as above, the N × M matrix ¯ ` = lim (z − z¯` )ρ(z) C ¯ = z→¯ z`

1 ¯ b(¯ z` )α(¯ ¯ z` ) a ¯0 (¯ z` )

(5.113)

¯ n (z). is the norming constant that relates the residues of µ ¯ n (z) to the function N − Note that if Cj denotes the norming constant at −zj , then C− j =

1 a0 (−z

j)

b(−zj )α(−zj ) =

1 −a0 (z

j)

(−b(zj ))α(zj ) = Cj .

(5.114)

Similarly, the norming constant associated with −¯ z` is equal to the norming constant associated with the eigenvalue z¯` . Hence, for any potentials Qn , Rn such that the Jost functions are welldefined we have estabilished the following result: Symmetry 5.1 All the eigenvalues appear in pairs ±zj (±¯ z` ). Moreover, the norming constants associated with −zj (respectively, −¯ z` ) is equal to the norming constant associated with +zj (respectively, +¯ z` ). 97

Symmetries In this section we compute symmetries of the scattering data which are induced by the symmetries in the potentials required to reduce discrete matrix NLS (5.1)–(5.2) to the system (1.9)–(1.10). As we have noted previously, these symmetries in the scattering data must be included in the formulation of the inverse problem in order to construct solutions. Recall that in order to reduce the discrete matrix NLS (5.1)–(5.2) to the discrete VNLS, we required two symmetries. First we impose the reduction (5.6) on the potentials, i.e. we restrict ourselves to potential matrices Qn , Rn such that Qn Rn = Rn Qn = αn I.

(5.115)

For N = 2, we obtain potentials which satisfy this symmetry by imposing the condition (5.8) or, equivalently, (5.9). Second, in order to reduce the system (1.9)–(1.10) to the single vector equation (5.7), we impose the further symmetry Rn = ∓QH n.

(5.116)

Each of the symmetries (5.115) and (5.116) induces a symmetry in the scattering data. We first consider the case when N = 2 and the potentials satisfy the symmetry (5.8) or, equivalently, (5.9). We will show how to compute the corresponding symmetries of norming constants and reflection coefficients. In order to compute the symmetry of the reflection coefficients, we define the functional iT h ¯n (i/z) φn (z) ˆ n−1 φ fn (z) = P

where the superscript

T

denotes standard matrix transposition and   (−i)n P 0 ˆ Pn = 0 in+1 P

¯n (z) are the solutions of the block-matrix scattering problem P is given by (5.10) and φn (z) and φ (5.20) defined by the boundary conditions (5.22). Note that this functional is well-defined for ¯n (z) exists for |z| ≤ 1. We relate this functional |z| ≥ 1 when (i) φn (z) exists for |z| ≥ 1 and (ii) φ to the reflection coefficients by comparing the limits limn→+∞ fn and limn→−∞ fn . First of all, we calculate the relation between fn and fn+1 by making use of the relations iT h ¯n+1 (i/z) φn+1 (z) ˆ nφ fn+1 (z) = P iT h ¯n (i/z) Sn (z)φn (z) ˆ n Sn (i/z)φ = P h iT h iT ¯n (i/z) ˆ n−1 φ ˆ n Sn (i/z)P ˆ −1 = P P Sn (z)φn (z) n−1 ¯n (z) satisfy the scattering where the second equality holds because, by definition, φn (z) and φ problem (5.20). The last expression can be simplified further. A direct calculation shows that h iT ˆ n Sn (i/z)P ˆ −1 P Sn (z) = (1 − αn ) I n−1

where, as we have assumed, Qn Rn = αn I. Hence, iT h ¯n (i/z) φn (z) = (1 − αn )fn (z) ˆ n−1 φ fn+1 (z) = (1 − αn ) P 98

(5.117)

and for any j ∈ N0 "n+j−1 #−1 Y fn (z) = (1 − αl ) fn+j (z)

(5.118)

l=n



fn (z) = 

n−1 Y

l=n−j



(1 − αl ) fn−j (z).

(5.119)

Now we evaluate the right-hand sides of (5.118)–(5.119) in the limit j → ∞. First, we rewrite the right-hand side of (5.118) by using the scattering equations (5.83)–(5.84) to obtain T ¯T ¯ ˆ Tn+j−1 N ¯ n+j (z)a(z) fn+j (z) = in+j b(i/z) Nn+j (i/z)P T ¯T ¯ ˆ Tn+j−1 Nn+j (z)b(z) +in+j z −2(n+j) b(i/z) Nn+j (i/z)P

ˆ Tn+j−1 N ¯ n+j (z)a(z) +i−(n+j) z 2(n+j) ¯ a(i/z)T NTn+j (i/z)P ˆ Tn+j−1 Nn+j (z)b(z) +i−(n+j) ¯ a(i/z)T NTn+j (i/z)P which, for |z| = 1, yields T ¯ lim fn+j (z) = −ib(i/z) P a(z) − ¯ a(i/z)T P b(z).

j→+∞

Hence, by (5.118), we have #−1 "+∞ Y   T ¯ −ib(i/z) P a(z) − ¯ a(i/z)T P b(z) . (1 − αl ) fn (z) =

(5.120)

l=n

On the other hand, ˆ Tn−j−1 φn−j (z) lim fn−j (z) = lim φ¯Tn−j (i/z)P

j→+∞

j→+∞

¯ Tn−j (i/z)P ˆ Tn−j−1 Mn−j (z) = lim ij−n z 2(n−j) M j→+∞

=0 so, by (5.119), we get fn (z) =

"

n−1 Y

#

(1 − αl ) 0 = 0.

l=−∞

(5.121)

Comparing (5.120) and (5.121), we obtain T ¯ −ib(i/z) P a(z) − ¯ a(i/z)T P b(z) = 0,

which, by definition (5.105) of the reflection coefficients, is equivalent to ρ(i/z) ¯ = −iP ρ(z)T P,

(5.122)

that is, if the potentials satisfy the symmetry (5.6), then the reflection coefficients satisfy (5.122). The symmetry (5.6) in the potentials also induces a symmetry in the eigenvalues. We now compute this symmetry. We have shown that the eigenvalues in |z| > 1 are the zeros of det a(z) 99

and the eigenvalues in |z| < 1 are the zeros of det ¯ a(z). To estabilish the relation between the eigenvalues in |z| > 1 and the eigenvalues in |z| < 1, we first estabilish a relationship between det a(z) and det c(z). Then we estabilish a relation between c(z) and ¯ a(i/z). Note that if |z| > 1, then |i/z| < 1. The equation (5.108) gives the relation between det a(z) and the Wronskian of φn (z), ψ n (z). With M = N and the additional symmetry (5.6), eq. (5.108) simplifies to  −N +∞ Y det a(z) (5.123) W (φn (z), ψ n (z)) =  (1 − αj ) j=n

where, as usual, αn IN = Qn Rn . On the other hand,     
d(z)
IN ¯ ¯ W (φn (z), ψ n (z)) = W , φn (z), φn (z) φn (z), φn (z) c(z) 0  N n−1 Y = (1 − αj ) det c(z)

(5.124)

j=−∞

where we used (5.82). Comparing (5.123) and (5.124) we obtain  −N +∞ Y det c(z) =  (1 − αj ) det a(z).

(5.125)

j=−∞

Analogously, the functional

h iT ˆ ¯n (i/z) ψ n (z) ˆ n−1 φ fn (z) = P

¯n (z) exists for |z| ≤ 1. By the is well-defined for |z| ≥ 1 when (i) ψ n (z) exists for |z| ≥ 1 and (ii) φ same argument that estabilished (5.117) for the functional fn (z), we obtain   +∞ Y ˆ f−∞ (z) f+∞ (z) =  (1 − αj ) ˆ j=−∞

where the subscript ±∞ is understood to mean the limit as n → ±∞. By evaluating the limits we obtain   +∞ Y aT (i/z) P =  ¯ (1 − αj ) P c(z). (5.126) j=−∞

Analogously, if we consider

iT h ˆ n−1 ψ ¯n (i/z) φn (z), ˆ n (z) = P h

which is well-defined for |z| ≥ 1, exactly the same arguments lead to the relation   +∞ Y c(i/z)T P P a(z) =  (1 − αj ) ¯ j=−∞

100

(5.127)

therefore 

+∞ Y



+∞ Y

det ¯ a(i/z) = 

j=−∞

det a(z) = 

j=−∞

Comparing (5.125) and (5.128) we have

M

(1 − αj )

det c(z)

N

(1 − αj ) det ¯ c(i/z).

(5.128)

(5.129)

det ¯ a(i/z) = det a(z), thus, zˆj = i/zj is an eigenvalue if, and only if, zj is an eigenvalue. Now we compute the symmetry of the norming constants associated with the pair of poles zj , zˆj = i/zj . We assume, as we have previously, that all the Jost functions are defined in the neighborhood of each eigenvalue. Let Cj be the norming constant associated with the eigenvalue ˆ j be the norming constant associated with the eigenvalue zˆj . The norming constants are zj and C given by (5.111) so ˆ j = lim (z − zˆj )¯ C ρ(z) = −izj−2 lim (w − zj ) ρ(i/w) ¯ w→zj

z→ˆ zj

and, applying the symmetry (5.122), we obtain ˆ j = −z −2 P CTj P. C j Hence, we have shown that: Symmetry 5.2 If the potentials Qn , Rn with N = 2 satisfy the symmetry (5.8) or, equivalently, (5.9) then: (i) the reflection coefficients satisfy the symmetry ρ(iz ¯ −1 ) = −iP ρT (z) P (ii) zˆj = i/zj is an eigenvalue such that |ˆ zj | < 1 if and only if zj is an eigenvalue such that |zj | > 1; (iii) the norming constants associated with these poles have the symmetry ˆ j = −z −2 P CTj P C j

(5.130)

ˆ j is the norming constant associwhere Cj is the norming constant associated with zj and C ated with zˆj . As a consequence, z˜j = i/¯ zj is an eigenvalue such that |˜ zj | > 1 if, and only if, z¯j is an eigenvalue such that |¯ zj | < 1 and the norming constants associated with these poles have the symmetry ¯ Tj P ˜ j = z¯−2 P C C j

(5.131)

¯ j is the norming constant associated to z¯j and C ˜ j is the norming constant associated where C with z˜j

101

¯ i.e. the number of eigenvalues inside the unit circle is equal to the number of eigenvalues and J = J, outside. Now we turn to the symmetry in the scattering data induced by the symmetry (5.116), i.e. Rn = ∓QH n , when the symmetry (5.115), Qn Rn = Rn Qn = αn IN , also holds. This symmetry in the scattering data is determined by methods similar to those above, but the calculation holds for any N . First, we consider the functional   ¯ (1/z ∗ ) H φ (z) gn± (z) = σ ± φ n n where

σ± =



IN 0 0 ±IM



.

(5.132)

Using the scattering problem (5.20), it is easy to show that when symmetries (5.115) and (5.116) hold, these functionals satisfy the recursion relation ± gn+1 (z) = (1 − αn )gn± (z).

(5.133)

Hence, 

± g+∞ (z) = 

+∞ Y

j=−∞



± (1 − αj ) g−∞ (z)

where the subscripts ±∞ are understood to denote the limits of gn± as n → ±∞. Evaluating these asymptotics by means of eqs. (5.22)–(5.23) and (5.83)–(5.84), we conclude that (for |z| = 1, or also |z| ≥ 1 provided the decay of the Jost functions is fast enough), the following relation holds ¯H (1/z ∗ )a(z) = 0 ±¯ aH (1/z ∗ )b(z) + b or equivalently, in terms of the reflection coefficients, ρ(z) ¯ = ∓ρH (1/z ∗ )

(5.134)

if symmetries (5.115) and (5.116) hold for the potentials. These symmetries in the potentials also induce a symmetry in the eigenvalues. Indeed, the func    ∗ H ¯n (1/z ∗ ) H ψ n (z) and ˆl± ¯ tionals g ˆn± (z) = σ ± φ φn (z) satisfy the recursion n (z) = σ ± ψ n (1/z ) relation (5.133) and the comparison between their asymptotic behaviors as n → ±∞ yields   +∞ Y (5.135) ¯ aH (1/z ∗ ) =  (1 − αj ) c(z) j=−∞



a(z) = 

+∞ Y

j=−∞

Finally, similar calculations show that



¯H (1/z ∗ ) =  b



cH (1/z ∗ ). (1 − αj ) ¯ +∞ Y

j=−∞

102



¯ (1 − αj ) d(z)

(5.136)

(5.137)



+∞ Y

bH (1/z ∗ ) = 

j=−∞

Note that (5.135) and (5.136) also imply  det ¯ a(1/z ∗) = 

+∞ Y

j=−∞



det a(z) = 

+∞ Y

j=−∞

which, taking into account (5.125), give  det ¯ a(1/z ∗ ) = 

+∞ Y

j=−∞



(1 − αj ) d(z). M

(1 − αj )

(5.138)

(det c(z))∗

N

∗

(1 − αj ) (det ¯ c(1/z ∗ )) M−N

(1 − αj )

∗

(det a(z)) .

(5.139)

We therefore conclude that z¯j = 1/zj∗ is an eigenvalue if, and only if, zj is an eigenvalue. The norming constant associated with the eigenvalue z¯j can be related to the norming constant associated with zj as follows: ¯ j = lim (z − z¯j ) ρ(z) C ¯ = ± lim z→¯ zj

w→zj

w∗ − zj∗ H ρ (w) = ±(zj∗ )−2 CH j. zj∗ w∗

We have therefore shown that: Symmetry 5.3 If the potentials satisfy the symmetry (5.116), i.e. Rn = ∓QH n , in addition to the symmetry (5.115), i.e. Qn Rn = Rn Qn = αn I, then: (i) the reflection coefficients satisfy the symmetry ρ(z) ¯ = ∓ρH (1/z ∗) (ii) z¯j = 1/zj∗ is an eigenvalue such that |¯ zj | < 1 if, and only if, zj is an eigenvalue such that |zj | > 1 and, consequently, the number of eigenvalues inside the unit circle is equal to the ¯ number of eigenvalues outside, i.e. J = J; (iii) the norming constants associated with these paired eigenvalues satisfy the symmetry ¯ j = ±(zj∗ )−2 CH C j

(5.140)

¯ j is the norming constant assowhere Cj is the norming constant associated with zj and C ciated with z¯j . As a consequence, if Symmetry 5.3 holds, then the eigenvalues appear in sets of eight J  ±zj , ±i/zj , ±1/zj∗, ±izj∗ j=1 .

Note that, since we have assumed that on the unit circle det a(z) 6= 0 and det ¯ a(z) 6= 0, from (5.91) we get the following relations ¯ a−1 (z)b(z) = ¯ a(z) − b(z)¯ c−1 (z) −1 ¯ ¯ a(z) − b(z)a (z)b(z) = c−1 (z) 103

(5.141) (5.142)

and then, taking into account the symmetries (5.135)–(5.136) and (5.137)–(5.138), we have +∞ h i Y −1 H ¯ a(z) − b(z)¯ a (z)b(z) a (z) = (1 − αj )IN

for |z| = 1.

(5.143)

for |z| = 1

(5.144)

j=−∞

Eq. (5.143) yields, from one side, 2



|det a(z)| = 

+∞ Y

j=−∞

N

−1 (1 − αj ) [det (IN − ρ(z)ρ(z))] ¯

and, from the other side, taking into account (5.136),

det



¯ ab b¯ a





=

+∞ Y

j=−∞

N

(1 − αj ) .

(5.145)

We can summarize the symmetries of the scattering data induced by the symmetries in the potentials as follows. In the general case, only Symmetry 5.1 holds, i.e. Symmetry 5.1 All the eigenvalues appear in pairs ±zj (±¯ z` ) outside and inside the unit circle respectively. Moreover, the norming constants associated with −zj (respectively, −¯ z` ) is equal to the norming constant associated with +zj (respectively, +¯ z` ). If the potentials Qn , Rn with N = 2 satisfy the symmetry (5.8), i.e. Rn = (−1)n PQTn P, then Symmetry 5.2 holds, i.e. Symmetry 5.2 (i) the reflection coefficients satisfy the symmetry ρ(iz ¯ −1 ) = −iP ρT (z) P (ii) zˆj = i/zj is an eigenvalue such that |ˆ zj | < 1 if and only if zj is an eigenvalue such that |zj | > 1; (iii) the norming constants associated with these poles have the symmetry ˆ j = −z −2 P CTj P C j ˆ j is the norming constant associwhere Cj is the norming constant associated with zj and C ated with zˆj . As a consequence, z˜j = i/¯ zj is an eigenvalue such that |˜ zj | > 1 if, and only if, z¯j is an eigenvalue such that |¯ zj | < 1 and the norming constants associated with these poles have the symmetry ˜ j = z¯−2 P C ¯ Tj P C j ¯ j is the norming constant associated to z¯j and C ˜ j is the norming constant associated where C with z˜j . Finally, if the potentials satisfy the symmetry Rn = ∓QH n , in addition to the symmetry Qn Rn = Rn Qn = αn I, then Symmetry 5.3 holds, i.e. Symmetry 5.3

104

(i) the reflection coefficients satisfy the symmetry ρ(z) ¯ = ∓ρH (1/z ∗) (ii) z¯j = 1/zj∗ is an eigenvalue such that |¯ zj | < 1 if, and only if, zj is an eigenvalue such that |zj | > 1; (iii) the norming constants associated with these paired eigenvalues satisfy the symmetry ¯ j = ±(zj∗ )−2 CH C j ¯ j is the norming constant assowhere Cj is the norming constant associated with zj and C ciated with z¯j . Trace formula J

Assume a(z) = det a(z) and a ¯(z) = det ¯ a(z) to have the simple zeros {±zj : |zj | > 1}j=1 and  J¯ ±¯ zj : |¯ zj | < 1 j=1 , respectively, and define J¯ ∗ −2 Y (z 2 − z¯m ) α ¯ (z) = a ¯(z). 2−z 2 ) (z ¯ m m=1

J ∗ −2 Y (z 2 − zm ) α(z) = a(z), 2 − z2 ) (z m m=1

(5.146)

According to (5.146), α(z) is analytic outside the unit circle, whereas it has no zeros, while α ¯ (z) is analytic inside, whereas it has no zeros; moreover, due to (5.100), α(z) → 1 as |z| → ∞. Therefore we can write   2 I J 2 X 1 log (¯ α(w)α(w)) z − zm − dw, |z| > 1 (5.147) log a(z) = log 2 − z∗ 2 z 2πi w−z |w|=1 m m=1   2 I J 2 X 1 log (α(w)¯ α(w)) z − z¯m dw |z| < 1. (5.148) + log a ¯(z) = log 2 ∗ 2 z − z¯m 2πi |w|=1 w−z m=1 If Symmetry 5.3 holds, the eigenvalues appear in sets of eight from (5.139) and (5.144) we obtain log a(z) =

J X

m=1

log

2 −2 (z 2 − zm )(z 2 + z¯m ) 1 + −2 2 )(z 2 + z 2πi (z 2 − z¯m ) m

I

|w|=1



±zj , ±i/zj , ±1/zj∗, ±izj∗

log [det (IN − ρ(w)ρ(w))] ¯ dw w−z



and

(5.149)

which allows one to recover a(z), a ¯(z) for any z in the proper region from knowledge of the discrete eigenvalues and of the reflection coefficient on the unit circle. The problem of reconstructing the matrices a(k) and ¯ a(k) is more complicated. However, from inverse scattering one can reconstruct potentials and eigenfunctions and then obtain the matrix transmission coefficients using, for instance, formulae (5.92) and (5.94).

5.2.2

Inverse Scattering Problem

Boundary Conditions and Residues In this section, we reconstruct the potentials from the scattering data. As in the 2 × 2 scattering problem, we first reconstruct the Jost functions from the scattering data and then recover the potentials from the Jost functions. 105

In order to formulate the inverse problem as a Riemann-Hilbert problem in the complex variable z, we must specify: (i) the boundary conditions of the Jost functions as |z| → ∞ and (ii) the equations that determine the residues of the poles of the meromorphic functions µn (z) and µ ¯n (z). According to the expansions (5.69), (5.76), and (5.100) and definition (5.102) we have the boundary conditions     0 0 , µ ¯n (z) → Nn (z) → IM ∆−1 n as |z| → ∞, with ∆n given by (5.77). Hence, the boundary condition for Nn (z) depends on Qn and Rn which are unknowns in the inverse problem. Therefore, we introduce the following modified functions    −1  −3 IN 0 −z Qn ∆−1 ) n + O(z N0n = Nn = (5.150) 0 ∆n IM + O(z −2 )     IN + O(z −2 ) IN 0 µn = µ0n = (5.151) z −1 ∆n Rn−1 + O(z −3 ) 0 ∆n     2 Ω−1 n + O(z ) ¯n = ¯ 0n = IN 0 N (5.152) N 3 0 ∆n −z∆n Rn Ω−1 n + O(z )     IN 0 zQn−1 + O(z 3 ) (5.153) µ ¯0n = µ ¯n = 0 ∆n ∆n + O(z 2 ) where Ωn is given by (5.79). Hence, by this modification of the Jost functions, we eliminate the dependence of the boundary conditions on the potentials. We emphasize that 0 does not indicate a derivative with respect to z, but rather the modification of the Jost functions by the matrix prefactor. From (5.103)–(5.104) it follows that these modified functions satisfy the jump conditions ¯ 0n (z) = z −2n N0n (z)ρ(z) µ0n (z) − N ¯ 0n (z)ρ(z). µ ¯0n (z) − N0n (z) = z 2n N ¯

(5.154) (5.155)

Also, the poles of µ0n (z) and µ ¯0n (z) are the same as the poles of µn (z) and µ ¯ n (z), respectively. The residues of these poles are determined by the relations: Res (µ0n ; zj ) = zj−2n N0n (zj )Cj ¯ 0n (¯ ¯j Res (¯ µ0n ; z¯j ) = z¯j2n N zj )C

(5.156) (5.157)

which follow from (5.110) and (5.112). We therefore solve the Riemann-Hilbert problem for the modified Jost functions (5.150)–(5.153). Recovery of the Jost functions ¯ 0n (z) from the scattering The first step of the inverse problem is to recover the functions N0n (z) and N data. Let us consider first the case when there are no discrete eigenvalues, i.e. µ0n and µ ¯ 0n have no poles. Introducing the (N + M ) × (N + M ) matrices
¯ 0n (z), µ mn (z) = (µ0n (z), N0n (z)) , m ¯ n (z) = N ¯ 0n (z) (5.158) with mn (z) analytic outside the unit circle |z| = 1 and m ¯ n (z) analytic inside, the “jump” conditions (5.154)–(5.155) can be written as mn (z) − m ¯ n (z) = m ¯ n (z)Vn (z) 106

|z| = 1

(5.159)

where −z 2n ρ(z) ¯ 0

−ρ(z)ρ(z) ¯ −2n z ρ(z)

Vn (z) =

!

(5.160)

and mn (z) → IN +M

as |z| → ∞.

(5.161)

Therefore (5.159) can be regarded as a generalized Riemann-Hilbert boundary value problem on |z| = 1 with boundary conditions given by (5.161). In analogy with what was done for the scalar case, we introduce the integral operators I 1 F(w) P¯ (F)(z) = lim dw (5.162) ζ→z 2πi |w|=1 w − ζ |ζ|<1

P (F)(z) = lim

ζ→z |ζ|>1

1 2πi

I

|w|=1

F(w) dw w−ζ

(5.163)

defined for |z| < 1 and |z| > 1, respectively, for any matrix-valued function F(w) continuous on |w| = 1. Applying P¯ to both sides of equations (5.159) yields I m ¯ n (w)Vn (w) 1 dw (5.164) m ¯ n (z) = IN +M − lim ζ→z 2πi |w|=1 w−ζ |ζ|<1

which allows one, in principle, to find m ¯ n (z) from the scattering data. In component form, (5.164) yields   I ¯ 0n (w)ρ(w)ρ(w) w−2n µ ¯ 0n (w)ρ(w) − N ¯ ¯ 0n (z) = IN − lim 1 N dw (5.165) 0 ζ→z 2πi |w|=1 w−ζ |ζ|<1

µ ¯0n (z) =



0 IM



+ lim

ζ→z |ζ|<1

1 2πi

I

|w|=1

¯ 0n (w)ρ(w) w2n N ¯ dw w−ζ

(5.166)

¯ 0n (z) and µ which is a system of linear integral equations, on |z| = 1, for N ¯ 0n (z) in terms of the scattering data. The solutions of these integral equations can be analytically continued into the region |z| < 1. Equivalently, applying the outside projector P to both sides of equations (5.154)–(5.155) we get   I ¯ 0n (w)ρ(w) 1 w2n N ¯ 0 0 dw + lim Nn (z) = IM ζ→z 2πi |w|=1 w−ζ |ζ|>1   I 1 w−2n N0n (w)ρ(w) IN 0 µn (z) = − lim dw. 0 ζ→z 2πi |w|=1 w−ζ |ζ|>1

To recover Qn and Rn we compute the power series expansions of the Jost functions around z = 0. By comparing the expansions (5.152) and (5.153) with the expansions about z = 0 of the right-hand sides of (5.165) and (5.166), or, equivalently, of (5.164), I z w−2 m ¯ n (w)Vn (w)dw + O(z −2 ) m ¯ n (z) = IN +M − 2πi |w|=1 107

we obtain I 1 ¯ 0(up) (w)ρ(w)dw Qn = w2n N ¯ n+1 2πi |w|=1 I   1 −2 ¯ 0(dn) w−2(n+1) µ ¯ 0(dn) (w)ρ(w) − w N (w) ρ(w)ρ(w) ¯ dw Rn = n n 2πi |w|=1 ¯ 0n (z) and µ where N ¯0n (z) are determined by the linear system (5.165)–(5.166). Hence, the formulation of the inverse problem is complete. The method of solution requires an extra step if µ0n (z) and µ ¯ 0n (z) have poles. Let us denote such poles as {zj }j=1,...J and {¯ zj }j=1,...J¯, respectively. We proceed as follows. First, we apply the inside projection operator (5.162) to both sides of (5.154) and the outside projection operator (5.163) to both sides of (5.155) to obtain ¯ 0n (z) = N



IN 0



J X

zj−2n



1 − lim ζ→z 2πi

I

+

j=1

|ζ|<1

N0n (z) =



0 IM



J¯ X

 1 1 0 0 N (zj ) + N (−zj ) Cj z − zj n z + zj n |w|=1

w−2n 0 N (w)ρ(w)dw w−ζ n

(5.167)



 1 ¯0 1 ¯0 ¯j Nn (¯ zj ) + Nn (−¯ zj ) C z − z¯j z + z¯j j=1 I w2n ¯ 0 1 Nn (w)ρ(w)dw ¯ + lim ζ→z 2πi |w|=1 w − ζ

+

z¯j2n

(5.168)

|ζ|>1

where N0n (zj ) is N0n (z) evaluated at the eigenvalue zj , N0n (−zj ) is N0n (z) evaluated at the com¯ 0n (¯ ¯ 0n (−¯ plementary eigenvalue −zj and similarly for N zj ) and N zj ). In expressions (5.167)–(5.168), we have taken into account Symmetry 5.1 by including poles in ± pairs with the appropriate norming constants. The eqs. (5.167)–(5.168) constitute a system of linear integral equations on |z| = 1. This system  0 J¯ J ¯ n (¯ ¯ 0n (−¯ zj ), N zj ) . We obtain and N depends on the additional matrices {N0n (zj ), N0n (−zj )} j=1

j=1

expressions for these matrices by evaluating (5.167) at the points ±¯ zj and (5.168) at the points ±zj . This results in a linear algebraic-integral system composed of (5.167)–(5.168) and ¯ 0n (¯ N zj ) =



IN 0



+



IN 0



zk−2n

k=1

−

¯ 0n (−¯ N zj ) =

J X

1 2πi

−



I

J X

|w|=1

zk−2n

k=1

1 − 2πi

I

 1 1 N0n (zk ) + N0n (−zk ) Ck z¯j − zk z¯j + zk w−2n 0 N (w)ρ(w)dw w − z¯j n



|w|=1

(5.169)

 1 1 0 0 N (zk ) + N (−zk ) Ck z¯j + zk n z¯j − zk n w−2n 0 N (w)ρ(w)dw w + z¯j n

108

(5.170)

N0n (zj )

=



0 IM



+



0 IM



z¯k2n

k=1

+

N0n (−zj ) =

J¯ X

1 2πi

−

I

J¯ X

1 2πi

 1 1 ¯ 0n (¯ ¯k N zk ) + N0n (−¯ zk ) C zj − z¯k zj + z¯k

|w|=1

z¯k2n

k=1

+





w2n ¯ 0 N (w)ρ(w)dw ¯ w − zj n

(5.171)

 1 1 ¯ 0n (¯ ¯k N zk ) + N0n (−¯ zk ) C zj + z¯k zj − z¯k

I

|w|=1

w2n ¯ 0 N (w)ρ(w)dw ¯ w + zj n

(5.172) J¯

where (5.169)–(5.170) hold for each eigenvalue {¯ zj : |¯ zj | < 1, Re z¯j ≥ 0}j=1 and (5.171)–(5.172) J

hold for each eigenvalue {zj : |zj | > 1, Re zj ≥ 0}j=1 . Note that in the system of equations (5.167)–(5.172), the matrix scattering data (i.e. ρ(z), ρ(z), ¯ Cj ¯ j ) always multiply on the right. This implies that the rows of the matrix equations are unand C coupled, i.e. the first row of (5.167) depends only on the first row of (5.169)–(5.170). In particular, 0(up) ¯ 0(up) ¯ 0(up) we consider the equations for the first N rows (i.e. the matrices N (z), Nn (z), N (¯ zj ) and n n 0(dn) ¯ 0(up) ¯ 0(dn) N (−¯ zj )) separately from the equations for the last M rows (i.e. the matrices N (z), Nn (z), n n 0(dn) 0(dn) ¯ n (¯ ¯ n (−¯ N zj ) and N zj )). The equations for the first N rows are consistent with the symmetry reductions ¯ 0(up) ¯ 0(up) N (−z) = N (z) n n

N0(up) (−z) = −N0(up) (z) n n

(5.173)

and the equations for the last M rows are consistent with the symmetry reductions ¯ 0(dn) ¯ 0(dn) (z) N (−z) = −N n n

(−z) = N0(dn) (z) N0(dn) n n

(5.174)

and these symmetries are consistent with the z-expansions we derived in the direct problem (cf. (5.76)–(5.78)). However, here the symmetry in the Jost functions is a consequence of the symmetry of the scattering data. By taking into account the symmetry reductions (5.173)–(5.174), we can split the system (5.167)–(5.172) into two smaller algebraic-integral systems. Under the symmetry reduction (5.173), the first N rows of (5.167)–(5.172) become the system ¯ 0(up) N (z) = IN + 2 n

I J X zj−2n+1 0(up) 1 w−2n 0(up) N (z )C − lim Nn (w)ρ(w)dw j j n 2 2 ζ→z 2πi |w|=1 w − ζ z − zj j=1

(5.175)

|ζ|<1

N0(up) (z) = 2 n

I J¯ X z¯j2n z w2n ¯ 0(up) 1 ¯ 0(up) ¯ N (¯ z ) C + lim Nn (w)ρ(w)dw ¯ j j n 2 2 ζ→z 2πi |w|=1 w − ζ z − z¯j j=1

(5.176)

|ζ|>1

¯ 0(up) N (¯ zj ) = IN + 2 n

I J X zk−2n+1 0(up) 1 w−2n 0(up) N (z )C − N (w)ρ(w)dw k k n z¯j2 − zk2 2πi |w|=1 w − z¯j n

(5.177)

k=1

N0(up) (zj ) n

I J¯ X 1 w2n ¯ 0(up) z¯k2n zj ¯ 0(up) ¯ N (w)ρ(w)dw ¯ N (¯ z ) C + =2 k k n zj2 − z¯k2 2πi |w|=1 w − zj n k=1

109

(5.178)

J¯

where (i) eq. (5.177) holds for each eigenvalue {¯ zj : |¯ zj | < 1, Re z¯j ≥ 0}j=1 and (ii) eq. (5.178) 0(up) J ¯ 0(up) holds for each eigenvalue {zj : |zj | > 1, Re zj ≥ 0} . The matrices N (−¯ zj ) and Nn (−zj ) n j=1

0(up) ¯ 0(up) are determined from, respectively, N (¯ zj ) and Nn (zj ), according to the symmetry
(5.173). n Hence, while the system (5.167)–(5.172) comprises equations corresponding to 2 J + J¯ eigenvalues, the system (5.175)–(5.178), which takes the symmetry (5.173) into account, comprises equations corresponding to only J + J¯ eigenvalues. Similarly, under the symmetry reduction (5.174), the last M rows of the system (5.167)–(5.172) become the system

¯ 0(dn) N (z) = 2 n

I J X zj−2n z 0(dn) 1 w−2n 0(dn) N (z )C − lim Nn (w)ρ(w)dw j j n ζ→z 2πi |w|=1 w − ζ z 2 − zj2 j=1

(5.179)

|ζ|<1

N0(dn) (z) n

= IM

I J¯ X z¯j2n+1 0(dn) w2n ¯ 0(dn) ¯ n (¯ ¯ j + lim 1 N z ) C +2 Nn (w)ρ(w)dw ¯ j ζ→z 2πi |w|=1 w − ζ z 2 − z¯j2 j=1

(5.180)

I J X zk−2n z¯j 0(dn) 1 w−2n 0(dn) N (z )C − N (w)ρ(w)dw k k z¯j2 − zk2 n 2πi |w|=1 w − z¯j n

(5.181)

|ζ|>1

¯ 0(dn) N (¯ zj ) = 2 n

k=1

N0(dn) (zj ) = IM + 2 n

I J¯ X z¯k2n+1 ¯ 0(dn) 1 w2n ¯ 0(dn) ¯ N (w)ρ(w)dw. ¯ N (¯ z ) C + k k n 2 2 zj − z¯k 2πi |w|=1 w − zj n

(5.182)

k=1

J¯

where (i) eq. (5.181) holds for each eigenvalue {¯ zj : |¯ zj | < 1, Re z¯j ≥ 0}j=1 and (ii) eq. (5.182) 0(dn) J ¯ 0(dn) holds for each eigenvalue {zj : |zj | > 1, Re zj ≥ 0} . The matrices N (−¯ zj ) and Nn (−zj ) n j=1

0(dn) ¯ 0(dn) are determined from, respectively, N (¯ zj ) and Nn (zj ) according to the symmetry (5.174). n Note that the system (5.175)–(5.178) is uncoupled from the system (5.179)–(5.182) and each is a closed system of algebraic-integral equations. These systems hold for general potentials. However, if the scattering data satisfy Symmetry 5.2, one can include this symmetry explicitly in the systems. Specifically, if Symmetry 5.2 holds, then the set of eigenvalues is given by J¯

J

{±¯ zj , ±i/¯ zj : |¯ zj | < 1, Re zj ≥ 0}j=1 ∪ {±zj , ±i/zj : |zj | > 1, Re z¯j ≥ 0}j=1

(5.183)

with the corresponding norming constants n

¯ j, C ˜j C

oJ¯

j=1

n oJ ˆj ∪ Cj , C

j=1

(5.184)

and therefore we can rewrite the system (5.175)–(5.176) as J¯ J X X (−1)n z¯j2n−1 0(up) zj−2n+1 0(up) ˜j N (z )C + 2i (i/¯ zj )C j j n −2 Nn 2 − z2 2+z z z ¯ j j j=1 j=1 I −2n w 1 N0(up) (w)ρ(w)dw − lim n ζ→z 2πi |w|=1 w − ζ

¯ 0(up) N (z) = IN + 2 n

|ζ|<1

110

(5.185)

N0(up) (z) n

J¯ J X X (−1)n zj−2n z 0(up) z¯j2n z ¯ 0(up) ¯ ¯ n (i/¯ ˆj =2 N zj )C N (¯ z ) C + 2 j j n 2 2−z 2 + z −2 z ¯ z j j j=1 j=1 I 2n 1 w ¯ 0(up) + lim N (w)ρ(w)dw ¯ n ζ→z 2πi |w|=1 w − ζ

(5.186)

|ζ|>1

J J¯ X X zk−2n+1 0(up) (−1)n z¯k2n−1 0(up) ˜k N (z )C + 2i (i/¯ zk )C k k n −2 Nn 2+z z¯j2 − zk2 z ¯ ¯ j k k=1 k=1 I w−2n 0(up) 1 N (w)ρ(w)dw (5.187) − 2πi |w|=1 w − z¯j n

¯ 0(up) N (¯ zj ) = IN + 2 n

¯ 0(up) N (i/zj ) = IN − 2 n 1 − 2π

N0(up) (zj ) n

N0(up) (i/¯ zj ) n

I

J J¯ X X zk−2n+1 0(up) (−1)n z¯k2n−1 0(up) ˜k N (z )C − 2i (i/¯ zk )C k k n −2 −2 −2 Nn 2 z + z z − z ¯ j j k k k=1 k=1

|w|=1

zj w−2n 0(up) N (w)ρ(w)dw 1 + izj w n

J J¯ X X (−1)n zk−2n zj ¯ 0(up) z¯k2n zj ¯ 0(up) ¯ ˆk Nn (i/zk )C N (¯ z ) C + 2 =2 k k 2 + z −2 zj2 − z¯k2 n z j k k=1 k=1 I w2n ¯ 0(up) 1 N (w)ρ(w)dw ¯ + 2πi |w|=1 w − zj n

(5.188)

(5.189)

J¯ X z¯k2n z¯j−1

J X (−1)n zk−2n z¯j−1 0(up) ¯ 0(up) ¯ ¯ n (i/zk )C ˆk N (¯ z ) C − 2i N k k n −2 −2 −2 2 + z ¯ − z z ¯ z ¯ j k k k=1 j k=1 I 1 z¯j w2n ¯ 0(up) + N (w)ρ(w)dw. ¯ (5.190) 2π |w|=1 1 + i¯ zj w n

= −2i

ˆ j is the norming constant associated with ±i/zj and C ˜ j is the norming constant Here, as before, C associated with ±i/¯ zj . Similarly, under this symmetry, the system (5.179)–(5.182) for the lower components of the eigenfunctions becomes J¯ J X X zj−2n z 0(dn) (−1)n z¯j2n z 0(dn) ˜j N (z )C + 2 (i/¯ zj )C j j n −2 Nn 2 − z2 2+z z z ¯ j j j=1 j=1 I −2n w 1 − lim N0(dn) (w)ρ(w)dw n ζ→z 2πi |w|=1 w − ζ

¯ 0(dn) N (z) = 2 n

(5.191)

|ζ|<1

N0(dn) (z) n

J J¯ X X (−1)n zj−2n−1 0(dn) z¯j2n+1 0(dn) ¯ n (¯ ¯ j + 2i ¯ n (i/zj )C ˆj N z ) C = IM + 2 N j 2 2−z 2 + z −2 z ¯ z j j j=1 j=1 I w2n ¯ 0(dn) 1 Nn (w)ρ(w)dw ¯ (5.192) + lim ζ→z 2πi |w|=1 w − ζ |ζ|>1

111

J J¯ X X zk−2n z¯j 0(dn) (−1)n z¯k2n z¯j 0(dn) ˜k (i/¯ zk )C N (z )C + 2 k k −2 Nn 2+z z¯j2 − zk2 n z ¯ ¯ j k k=1 k=1 I 1 w−2n 0(dn) − (w)ρ(w)dw N 2πi |w|=1 w − z¯j n

¯ 0(dn) N (¯ zj ) = 2 n

¯ 0(dn) N (i/zj ) = −2i n

J X zk−2n zj−1

N0(dn) (zk )Ck − 2i n 2

J¯ X (−1)n z¯k2n zj−1

z −2 + zk zj−2 − z¯k−2 k=1 k=1 j I zj w−2n 0(dn) 1 N (w)ρ(w)dw − 2π |w|=1 1 + izj w n

N0(dn) (zj ) n

(5.193)

˜k N0(dn) (i/¯ zk )C n (5.194)

J¯ J X X z¯k2n+1 ¯ 0(dn) (−1)n zk−2n−1 ¯ 0(dn) ˆk ¯ Nn (i/zk )C = IM + 2 N (¯ z ) C + 2i k k 2 + z −2 zj2 − z¯k2 n z j k k=1 k=1 I 1 w2n ¯ 0(dn) N (w)ρ(w)dw ¯ (5.195) + 2πi |w|=1 w − zj n

N0(dn) (i/¯ zj ) = IM − 2 n +

1 2π

I

J¯ J X X z¯k2n+1 ¯ 0(dn) (−1)n zk−2n−1 ¯ 0(dn) ¯ ˆk N (¯ z ) C − 2i Nn (i/zk )C k k n −2 2 z¯ + z¯k z¯j−2 − zk−2 k=1 j k=1

|w|=1

z¯j w2n ¯ 0(dn) N (w)ρ(w)dw ¯ 1 + i¯ zj w n

(5.196)

where the eigenvalues are given by (5.183). The linear algebraic-integral systems (5.185)–(5.190) and (5.191)–(5.196) determine the Jost ¯ 0n (z) and N0n (z). However, these linear systems do not account for Symmetry 5.3 functions N which relates the eigenvalues in the region |z| > 1 with the eigenvalues in the region |z| < 1. Note that we have not proved that the linear systems have a solution in general. Neither have we estabilished conditions on the scattering data which would ensure that these systems have a solution. Rather, we have shown how to obtain a linear algebraic-integral system for the Jost functions from the scattering data. Below, we compute the solution of (5.185)–(5.190) for a reflectionless potential (i.e. ρ(z) = ρ(z) ¯ = 0 on |z| = 1) with a single octet of eigenvalues {±z1 , ±i/z1, ±¯ z1 , ±i/¯ z1 : |z1 | > 1, |¯ z1 | < 1}. However, here we do not attempt to rigorously estabilish conditions under which either systems are guaranteed to have (unique) solutions. Recovery of the Potential We show how to recover the potentials Qn and Rn from the Jost functions and the scattering data. First, we show how to recover the matrix potential Qn . Then, we derive two methods for recovering the matrix potential Rn - one method that is applicable for general Rn and a simplified method for potentials that satisfy the symmetry Rn Qn = Qn Rn = αn I (cf. Symmetry 5.2). To recover Qn , we first apply the inside projection operator (5.162) to both sides of the jump condition (5.155). This yields the relations µ ¯ 0(up) (z) n

I J¯ X z¯j2n z 1 w2n ¯ 0(up) 0(up) ¯ ¯ =2 N (¯ z ) C + N (w)ρ(w)dw ¯ j j n 2 z 2 − z¯j 2πi |w|=1 w − z n j=1 112

(5.197)

µ ¯0(dn) (z) n

= IM

I J¯ X z¯j2n+1 0(dn) w2n ¯ 0(dn) ¯ n (¯ ¯j + 1 +2 N z ) C N (w)ρ(w)dw ¯ j z 2 − z¯j2 2πi |w|=1 w − z n j=1

(5.198)

when the symmetries (5.173)–(5.174) are taken into account. Now, by comparing the power series expansion (in z) of the right-hand side of (5.197) and the expansion (5.153) of µ ¯ 0n (z) we obtain Qn−1 = −2

J¯ X

2(n−1) ¯ 0(up) ¯j Nn (¯ zj )C

z¯j

+

j=1

1 2πi

I

|w|=1

¯ 0(up) w2(n−1) N (w)ρ(w)dw ¯ n

(5.199)

¯ 0(up) which gives the potential in terms of the Jost function N (z) evaluated on |z| = 1 and at the n ¯ J eigenvalues {±¯ zj : |¯ zj | < 1, Re z¯j > 0}j=1 . 0(dn) ¯ 0(dn) Analogously, we can recover Rn from the Jost functions Nn (z) and N (z). Indeed, by n applying the outside projection operator (5.163) to the jump condition (5.154), we obtain the equation µ0(dn) (z) = 2 n

I J X zj−2n z 0(dn) 1 w−2n 0(dn) N (z )C − N (w)ρ(w)dw. j j n 2 2 z − zj 2πi |w|=1 w − z n j=1

Then, by comparing the Laurent expansion (in z) of the right-hand side of this expression with the expansion (5.151) of µ0n (z), we obtain ∆n Rn−1 = 2

J X

zj−2n N0(dn) (zj )Cj + n

j=1

1 2πi

I

|w|=1

w−2n N0(dn) (w)ρ(w)dw. n

(5.200)

Similarly, by comparing the power series expansion (in z) of the right-hand side of (5.198) and the expansion (5.153) of µ ¯0n (z), we obtain ∆n = IM − 2

J¯ X

¯ 0(dn) ¯j (¯ zj )C z¯j2n−1 N n

j=1

1 + 2πi

I

|w|=1

¯ 0(dn) w2n−1 N (w)ρ(w)dw. ¯ n

(5.201)

Hence, with (5.200) and (5.201) we can recover Rn from the Jost functions evaluated on |z| = 1 and J J¯ at their respective eigenvalues {±zj : |zj | > 1, Re zj > 0}j=1 and {±¯ zj : |¯ zj | < 1, Re z¯j > 0}j=1 . If we explicitly include the effect of Symmetry 5.2 on the eigenvalues, then (5.199) becomes Qn−1 = −2

J¯ X

2(n−1) ¯ 0(up) ¯j zj )C Nn (¯

z¯j

−2

−2(n−1) ¯ 0(up) ˆj Nn (i/zj )C

(−1)n−1 zj

j=1

j=1

1 + 2πi

J X

I

|w|=1

¯ 0(up) w2(n−1) N (w)ρ(w)dw ¯ n

(5.202)

where the eigenvalues are as in (5.183). Also, there is a simpler procedure for recovering the potential Rn when the potentials satisfy 0(dn) ¯ 0(dn) the symmetry (5.115). Recall that the Jost functions Nn (z) and N (z) satisfy (5.179) and n (5.180). Comparing the power series expansion (in z) of the right-hand side of (5.179) with the ¯ 0(dn) expansion (5.152) of N (z) yields n ∆n Rn Ω−1 n

=2

J X j=1

−2(n+1) 0(dn) zj Nn (zj )Cj

1 + 2πi

113

I

|w|=1

w−2(n+1) N0(dn) (w)ρ(w)dw. n

In general, this relation is insufficient to recover Rn as we must also determine ∆n and Ω−1 n . However, if the potentials satisfy (5.115), then Ωn = ∆ n =

+∞ Y

(1 − αk ) I

(5.203)

k=n

and therefore Rn = 2

J X

−2(n+1)

zj

N0(dn) (zj )Cj +2 n

j=1

J¯ X 2(n+1) 0(dn) ˜j (−1)n+1 z¯j Nn (i/¯ zj )C j=1

+

1 2πi

I

|w|=1

w−2(n+1) N0(dn) (w)ρ(w)dw. n

(5.204)

Note that the symmetry (5.122) for the reflection coefficients can be obtained from the inverse problem formulae (5.202) and (5.204) reconstructing the potentials in terms of the scattering data. Indeed, let us restrict ourselves for simplicity to the case N = 2 and consider “small” potentials Qn and Rn satifying the symmetry Rn = (−1)n PQTn P.

(5.205)

Moreover, let us assume no discrete eigenvalues. Hence, due to the small norm assumption for ¯ (up) (w) ∼ IN and N0(dn) (w) ∼ IM and the potentials, from (5.185) and (5.191) it follows that N0 substitution into (5.202)–(5.204) yields I 1 w2n ρ(w)dw ¯ (5.206) Qn ∼ 2πi |w|=1 I 1 Rn ∼ w−2(n+1) ρ(w)dw. (5.207) 2πi |w|=1 Performing in (5.206) the change of variable w → i/w and taking into account that the potentials satifsy (5.205), immediatly yields the symmetry (5.122) Reflectionless Potentials In this section, we consider the recovery of potentials from scattering data such that ρ(z) = ρ(z) ¯ = 0 on |z| = 1. We refer to such potentials as reflectionless potentials. In particular, we consider scattering data that satisfy Symmetry 5.2 and Symmetry 5.3. We explicitly calculate the potential in the case where there is one octet of eigenvalues and norming contants. In this case we obtain potentials with the familiar sech profile with a complex modulation. However, in order to obtain this potential, we impose an additional condition on the norming constants. If ρ(z) = ρ(z) ¯ = 0 on |z| = 1, then the integrals vanish in the algebraic-integral system (5.185)–(5.190). In this case, this system reduces to the linear algebraic system ¯ 0(up) N (¯ zj ) = IN + 2 n

¯ 0(up) N (i/zj ) = IN − 2 n

J J¯ X X zk−2n+1 0(up) (−1)n z¯k2n−1 0(up) ˜k (i/¯ zk )C N (z )C + 2i k k n −2 Nn 2+z z¯j2 − zk2 z ¯ ¯ j k k=1 k=1

(5.208)

J J¯ X X zk−2n+1 0(up) (−1)n z¯k2n−1 0(up) ˜k N (z )C − 2i (i/¯ zk )C k k n −2 −2 −2 Nn 2 z + z z − z ¯ j j k k k=1 k=1

(5.209)

114

N0(up) (zj ) n

J¯ J X X (−1)n zk−2n zj ¯ 0(up) z¯k2n zj ¯ 0(up) ˆk ¯ =2 N (¯ z ) C + 2 Nn (i/zk )C k k zj2 − z¯k2 n zj2 + zk−2 k=1

N0(up) (i/¯ zj ) n

= −2i

(5.210)

k=1

J¯ X z¯k2n z¯j−1

J X (−1)n zk−2n z¯j−1 0(up) ¯ 0(up) ¯ k − 2i ¯ n (i/zk )C ˆ k. N (¯ z ) C N k n −2 −2 −2 2 z ¯ + z ¯ z ¯ − z j j k k k=1 k=1

(5.211)

Moreover, the expression (5.202) for the potential Qn reduces to Qn−1 = −2

J¯ X

2(n−1) ¯ 0(up) ¯j Nn (¯ zj )C

z¯j

−2

J X

−2(n−1) ¯ 0(up) ˆ j. Nn (i/zj )C

(−1)n−1 zj

(5.212)

j=1

j=1

Analogously, the system (5.193)–(5.196) reduces to the linear algebraic system ¯ 0(dn) N (¯ zj ) = 2 n

J J¯ X X zk−2n z¯j 0(dn) (−1)n z¯k2n z¯j 0(dn) ˜k Nn (i/¯ zk )C N (z )C + 2 k k z¯j2 − zk2 n z¯j2 + z¯k−2 k=1

¯ 0(dn) N (i/zj ) = −2i n

J X zk−2n zj−1

(5.214)

J¯ J X X (−1)n zk−2n−1 ¯ 0(dn) z¯k2n+1 ¯ 0(dn) ¯ ˆk Nn (i/zk )C N (¯ z ) C + 2i k k n 2 2 zj − z¯k zj2 + zk−2

(5.215)

J J¯ X X (−1)n zk−2n−1 ¯ 0(dn) z¯k2n+1 ¯ 0(dn) ¯ ˆk N (¯ z ) C − 2i Nn (i/zk )C k k n z¯−2 + z¯k2 z¯j−2 − zk−2 k=1 k=1 j

(5.216)

N0(dn) (zk )Ck n zk2

− 2i

k=1

zj−2 − z¯k−2

k=1

k=1

N0(dn) (i/¯ zj ) = IM − 2 n

J¯ X (−1)n z¯k2n zj−1

˜k N0(dn) (i/¯ zk )C n

z −2 + k=1 j

N0(dn) (zj ) = IM + 2 n

(5.213)

k=1

and (5.204) gives the reconstruction of the potential Rn Rn = 2

J X

−2(n+1) 0(dn) zj Nn (zj )Cj

j=1

+2

J¯ X

2(n+1)

(−1)n+1 z¯j

˜j N0(dn) (i/¯ zj )C n

(5.217)

j=1

Therefore, in the reflectionless case (i.e. when ρ(z) = ρ(z) ¯ = 0 on |z| = 1) and with Symmetry ¯ 5.2 taken into account, the potentials are determined by the solutions of the 2(J + J)-dimensional linear algebraic systems (5.208)–(5.211) and (5.213)–(5.216). In particular, if there is one octet of eigenvalues and norming constants, i.e. o   n
 ˜1 , ˆ 1 , ±i/¯ ¯ 1 , ±i/z1 , C (5.218) z1 , C (±z1 , C1 ) , ±¯ z1 , C the system (5.208)–(5.211) reduces to

"

# 2n −2(n−1) n 4n+2 z ¯ z (−1) z ¯ 1 ¯ 0(up) ¯ 0(up) ¯ 1C ˜1 ¯ 1 C1 − N (¯ z1 ) = IN − 4N (¯ z1 ) 1 2 1 2 2 C C n n (¯ z1 − z1 ) (1 + z¯14 )2 # " −2(n−1) 2(n+1) n −4(n−1) z z ¯ (−1) z 1 1 0(up) 1 ¯ n (i/z1 ) ˆ 1 C1 + ˆ 1C ˜ 1 (5.219) −4N C C (z12 − z¯12 )(1 + z14 ) (1 + z¯14 )(¯ z12 − z12 ) 115

"

# n 2 4n+2 2n −2(n−2) (−1) z z ¯ z ¯ z 1 1 1 1 ¯ 0(up) ¯ 0(up) ¯ 1 C1 − ¯ 1C ˜1 N (i/z1 ) = IN − 4N (¯ z1 ) C C n n (z12 − z¯12 )(1 + z14 ) (1 + z¯14 )(z12 − z¯12 ) # " −2(n−2) 2(n+1) z1 z¯1 (−1)n z1−4n+6 ˆ 0(up) ˆ ˜ ¯ C1 C1 − C1 C1 (5.220) −4Nn (i/z1 ) (1 + z14 )2 (z12 − z¯12 )2 0(up)

0(up)

where we have eliminated the matrices Nn (z1 ) and Nn (i/¯ z1 ). So far, in deriving the system (5.219)–(5.220), we have accounted for the effect of Symmetry 5.2 on the eigenvalues, but we have not considered the effect of this symmetry on the norming constants. We now use the symmetry in the norming constants to further simplify the solution of this system. For N = 2, we define ! ! (1) (2) (1) (2) γ¯1 γ¯1 γ1 δ1 2 ¯ . (5.221) , C1 = z¯1 C1 = (2) (1) (2) (1) δ¯1 −δ¯1 γ1 −δ1 Then, the symmetries in the norming constants given by Symmetry 5.2, namely, the relations (5.130)–(5.131), imply that ! ! (1) (2) ¯(1) γ¯ (2) δ δ δ −2 1 1 1 1 ˜1 = ˆ 1 = −z , C (5.222) C 1 (2) (1) (2) (1) γ1 −γ1 δ¯1 −¯ γ1 and correspondingly, ˆ 1 C1 = C1 C ˆ 1 = −z −2 (γ1 · δ1 ) I C 1
˜ 1 = z¯12 ¯ ˜ 1C ¯1 = C ¯ 1C γ1 · δ¯1 I C

where (1)

γ1 =

γ1 (2) γ1

!

(1)

,

δ1 =

δ1 (2) δ1

!

(1)

,

γ¯1 (2) γ¯1

γ1 = ¯

!

(5.223) (5.224)

¯ δ1 =

,

(1) δ¯1 (2) δ¯1

!

.

(5.225)

If (as explained below) we impose the additional condition δ1 = δ¯1 = 0

(5.226)

then the system (5.219)–(5.220) has the unique solution ¯ 0(up) N (¯ z1 ) = n ¯ 0(up) N (iz1−1 ) = n

1 1+gn

0 1−

0 1−

(z12 −¯z12 )gn z12 (1+¯ z14 )(1+gn )

z12 (z12 −¯ z12 )gn (1+z14 )(1+gn )

0

0 1 1+gn

!

!

(5.227)

(5.228)

where gn = 4 (γ1 · ¯ γ1 ) z12 − z¯12


−2

116

−2(n−1) 2(n+1) z¯1 .

z1

(5.229)

Substituting these expressions into (5.210)–(5.211) yields   2(n+1) 2(n+1) (2) z¯1 z1 (1) z¯1 z1 γ ¯ γ ¯ 2  1 1 z12 −¯ z12 z12 −¯ z12  N0(up) (z1 ) = −2n−1 −2n−1 n n (1) z1 1 + gn (−1)n+1 γ (2) z21 −2 (−1) γ1 z2 +z−2 1 z +z 1 1 1 1   (1) z¯12n+1 (2) z¯12n+1 γ1 z¯−2 +¯z2 ¯ γ1 z¯−2 −¯z2 ¯ −2i 1 1 1 1   N0(up) (i¯ z1−1 ) = −2(n+1) −1 −2(n+1) −1 n z¯1 z¯1 n (1) z1 1 + gn (−1)n+1 γ (2) z1 −2 −2 (−1) γ 1 1 z¯ −z z¯−2 −z −2 1

1

1

1

so that finally



¯ 0(up) N (z) =  n

1 + 4 kγ1 k

2

2(n+1) −2(n−1) z¯1 z1 (z 2 −z12 )(z12 −¯ z12 )(1+gn )

0

0 1 − 4 kγ1 k2

2(n−1) −2(n+1)

z¯1 z1 (z 2 +¯ z1−2 )(¯ z1−2 −z1−2 )(1+gn )



.

(5.230)

By (5.202), taking into account (5.8) and (5.212), we obtain for the potentials the following expressions ! ! 2(n+1) (1) (1) z¯1 Qn γ¯1 = −2 (5.231) (2) (2) −2 2(n+2) Qn γ¯1 1 + 4 (γ1 · ¯ γ1 ) (z12 − z¯12 ) z1−2n z¯1 ! ! −2(n+1) (1) (1) z1 Rn γ1 =2 . (5.232) (2) (2) −2 2(n+2) Rn γ1 1 + 4 (γ1 · ¯ γ1 ) (z12 − z¯12 ) z1−2n z¯1 We now consider the effect of Symmetry 5.3 on the inverse problem. Specifically, we restrict ourselves to the case Qn = −RH n , since it is known that in the defocusing case no discrete eigenvalues are allowed for decaying potentials. Recall that if Symmetry 5.3 holds, then J¯ = J and z¯j = 1/zj∗ for any j = 1, . . . , J. Hence, in this case, the eigenvalue/norming constants octet (5.218) takes the form n   o
 ¯ 1 , ±i/z1 , C ˆ 1 , ±iz1∗, C ˜1 . (±z1 , C1 ) , ±1/z1∗, C (5.233)

Moreover, in terms of the norming costants, Symmetry 5.3 implies that ¯ γ1 = γ1∗ in (5.221) and (5.222). With this symmetries explicitly included, as well as the substitution z1 = ea1 +ib1 , the potentials (5.231)–(5.232) are ! (1) γ∗ Qn (5.234) = − 1 e2ib1 (n+1) sinh(2a1 )sech (2a1 (n + 1) + d) (2) kγ1 k Qn ! (1) γ1 −2ib1 (n+1) Rn e sinh(2a1 )sech (2a1 (n + 1) + d) (5.235) = (2) kγ1 k Rn where d = log [sinh(2a1 )] − log kγ1 k. These potentials are a discrete, vector version of the familiar sech profile with complex modulation. If, instead of the condition (5.226), i.e. δ1 = δ¯1 = 0, we choose γ1 = ¯ γ1 = 0

117

(5.236)

then, according to (5.8) and (5.212), the solution of the system (5.219)–(5.220) yields the potentials ! ! −2(n+1) (1) (1) (−1)n z1 δ1 Qn (5.237) =2
(2) (2) −2 2(n+2) δ1 Qn 1 + 4 δ1 · δ¯1 (z12 − z¯12 ) z1−2n z¯1 ! ! 2(n+1) (1) (1) (−1)n z¯1 δ¯1 Rn . (5.238) = −2
(2) (2) −2 2(n+2) δ¯ Rn 1 + 4 δ · δ¯ (z 2 − z¯2 ) z −2n z¯ 1

1

1

1

1

1

1

If Symmetry 5.3 holds, then δ¯1 = δ1∗ and, as before, z¯1 = 1/z1∗ and with this symmetry and the substitution z1 = ea1 +ib1 , the potentials (5.237)–(5.238) are ! (1) δ Qn = − 1 e−2i(b1 +π/2)(n+1) sinh(2a1 )sech (2a1 (n + 1) + d) (5.239) (2) kδ Qn 1k ! (1) δ∗ Rn = 1 e2i(b1 +π/2)(n+1) sinh(2a1 )sech (2a1 (n + 1) + d) (5.240) (2) kδ1 k Rn where, here, d = log [sinh(2a1 )] − log kδ1 k. Like the potentials (5.234)–(5.235), the potentials (5.239)–(5.240) are a discrete version of the familiar sech profile with complex modulation. Note that, in order to obtain the above complex-modulated sech potential, we imposed the condition (5.226) or (5.236) on the norming constants. However, even in the case of a single octet of eigenvalues/norming constants, there exists a more general solution of the system (5.219)– (5.220) corresponding to the case when we do not restrict the norming constants to obey the condition γ1 = 0 or δ1 = 0. The general case γ1 6= 0, δ1 6= 0 corresponds to solutions that can be called “composit” solitons, as opposite to the “fundamental” solitons we obtain for γ1 = 0 or δ1 = 0. Even though a single octect is apparently the minimal number of discrete eigenvalues, the composite solitons essentially differ from the fundamental solitons. In fact, one composite soliton could be regarded as a two-soliton solution (it has two peaks, instead of only one). The special case γ1 6= 0, δ1 6= 0 but γ1 · δ1 = 0 is still a composite soliton but such that the two peaks coincide. We can also write down the solution of the linear system (5.219)–(5.220) when γ1 and δ1 are possibly both different from zero but such that γ1 · δ1 = 0. In this case one obtains " # 2 −2  2 ∗ 1 g ˜ + z z ¯ kγ k W (γ , δ ) n 1 1 1 0(up) 1 1 ¯ n (i/z1 ) = N I+ 2 2 −2 1 + g˜n W (γ1 , δ1∗ ) kδ1 k2 kγ1 k + kδ1 k z12 + z1 "  # z¯12 + z1−2 g˜n 1 kδ1 k2 −W (γ1∗ , δ1 ) 0(up) ¯ I+ Nn (¯ z1 ) = 2 −2 2 2 1 + g˜n −W (γ1 , δ1∗ ) kγ1 k kγ1 k + kδ1 k z¯12 + z¯1 where g˜n = 4

−2(n−1) 2(n+1) z¯1 2 2 (z1 − z¯12 )

z1

and, as usual,



(1) (2)

2

kγ1 k + kδ1 k

2



(2) (1)

W (γ1 , δ1 ) = γ1 δ1 − γ1 δ1 .

(5.241)

(5.242)

Hence (5.231)–(5.232) yield (1)

Qn (2) Qn (1)

Rn (2) Rn

!

= −2

!

=2

2(n+1)

−2(n+1)

z¯1 n z γ ∗ + 2 (−1) 1 δ1 1 + g˜n+1 1 1 + g˜n+1

−2(n+1)

(5.243)

2(n+1)

z1 z¯ γ1 + 2 (−1)n 1 δ∗. 1 + g˜n+1 1 + g˜n+1 1 118

(5.244)

Taking into account Symmetry 5.3 and substituting z1 = ea1 +ib1 yields ! (1) i h sinh 2a1 Qn iπ+2i(n+1)b1 ∗ γ1 + e−2i(n+1)b1 −inπ δ1 . = 1/2 sech (2(n + 1)a1 + d) e (2) Qn 2 2 kγ1 k + kδ1 k

(5.245)

1 2 2 log (kγ1 k + kδ1 k ) 2

d = log (sinh 2a1 ) −

(5.246)

Gel’fand-Levitan- Marchenko integral equations Like in the scalar case, we can also provide a reconstruction for the potentials by means of Gel’fand¯n in Levitan-Marchenko integral equations. Indeed, let us represent the eigenfunctions ψ n and ψ terms of triangular kernels ψ n (z) =

+∞ X

z −j K(n, j)

|z| > 1

(5.247)

j=n

¯n (z) = ψ

+∞ X

¯ z j K(n, j)

|z| < 1

(5.248)

j=n

where K(n, j) =



K(up) (n, j) K(dn) (n, j)



¯ K(n, j) =

,



¯ (up) (n, j) K ¯ (dn) (n, j) K



(5.249)

and write the equations (5.83)–(5.84) in the form ¯n (z) = ψ n (z)ρ(z) φn (z)a−1 (z) − ψ −1 ¯n (z)ρ(z). ¯ φn (z)¯ a (z) − ψ n (z) = ψ ¯

(5.250) (5.251)

H 1 −m−1 Applying the operator 2πi for m ≥ n to the equation (5.250) and taking into |z|=1 dz z account the asymptotics (5.69)–(5.70), (5.76), (5.78) and (5.100)–(5.101), as well as the triangular representations (5.247)–(5.248), we obtain ¯ K(n, m) +

+∞ X

K(n, j)F(m + j) =

j=n



IN 0



δm,n

m≥n

(5.252)

where F(n) =

J X

1 2πi

zj−n−1 Cj +

j=1

Analogously, operating on eq. (5.251) with K(n, m) +

+∞ X

1 2πi

H

|z|=1

I

119

(5.253)

dz z −m−1 for m ≥ n yields

¯ ¯ K(n, j)F(m + j) =

j=n

z −n−1 ρ(z)dz.

|z|=1



0 IM



δm,n

m≥n

(5.254)

where ¯ F(n) =−

J¯ X

¯j z¯jn−1 C

j=1

1 + 2πi

I

z n−1 ρ(z)dz. ¯

(5.255)

|z|=1

Eqs. (5.252) and (5.254) constitute the Gel’fand-Levitan-Marchenko equations. Note that the sum into (5.253) (resp. (5.255)) is performed over all the discrete eigenvalues which are inside (resp. outside) the unit circle. Since these eigenvalues are paired and the corresponding norming constants satisfy (5.114), the GLM equations can be simplified as follows +∞ X

¯ K(n, m) +

K(n, j)FR (m + j) =



IN 0



δm,n

m≥n

(5.256)

¯ ¯ R (m + j) = K(n, j)F



0 IM



δm,n

m≥n

(5.257)

n = odd n = even

(5.258)

j=n j+m=odd +∞ X

K(n, m) +

j=n j+m=odd

where  PJ R 1 2 j=1 zj−n−1 Cj + πi z −n−1 ρ(z)dz CR FR (n) = 0 ( R PJ¯ n−1 ¯j + 1 ρ(z)dz ¯ −2 j=1 z¯jn−1 C ¯ πi CR z FR (n) = 0

n = odd n = even

(5.259)

and CR denotes the right half of the unit circle. Comparing the representations (5.247)–(5.248) for the eigenfuctions with the asymptotics (5.76) and (5.78) and recalling (5.25), yields the reconstruction of the potentials in terms of the kernels of GLM equations, i.e. ¯ (dn) (n, n) = 0, K(up) (n, n) = K

¯ (up) (n, n) = Ω−1 K n ,

K(dn) (n, n) = ∆−1 n

¯ (dn) (n, n + 1)K ¯ (up) (n, n)−1 . Rn = −K

Qn = −K(up) (n, n + 1)K(dn) (n, n)−1 ,

(5.260) (5.261)

If the potentials satisfy Symmetry 5.2, we define for m > n K(n, m) = ∆−1 n κ(n, m) ¯ K(n, m) = Ω−1 ¯ (n, m) n κ

(5.262) (5.263)

and κ(n, n) =



0 IM



κ ¯ (n, n) =



IN 0



.

(5.264)

κ(n, j)FR (m + j) = 0

(5.265)

¯ R (m + j) = 0 κ ¯ (n, j)F

(5.266)

Eqs. (5.256) and (5.257) then yield for m > n ¯ m) + k(n,

κ(n, m) +



0 IM



FR (n + m) +



IN 0



¯ R (n + m) + F

+∞ X

j=n j+m=odd +∞ X

j=n j+m=odd

120

Q∞ and since ∆n = Ωn = k=n (1 − αk )I, from (5.260) and (5.261) it follows that κ and κ ¯ are related to the potentials as follows Qn = −κ(up) (n, n + 1)

5.2.3

Rn = −¯ κ(dn) (n, n + 1).

(5.267)

Time evolution

The time evolution of the scattering data is obtained in the same way as for the scalar case. The operator (5.4) determines the evolution of the Jost functions. From this we deduce the time evolution of the scattering data. Since we have assumed that Qn , Rn → 0 as n → ±∞, then the time dependence (5.4) is asymptotically of the form   −iµIN − iA 0 ∂τ vn = vn as n → ±∞ (5.268) 0 iµIM + iB where µ=


2 1 z − z −1 . 2

The constant matrices A and B can be absorbed by a gauge transformation (5.5), therefore, in order to simplify the calculation, we determine the time-dependence of the scattering data in the gauge A, B = 0. In this case, the system (5.268) has solutions that are linear combinations of the solutions     IN 0 . vn+ = e−iµτ , vn− = eiµτ IM 0 However, such solutions are not compatible with the fixed boundary conditions of the Jost functions (5.24)–(5.25) and therefore we define the time-dependent functions Mn (z, τ ) = e−iµτ Mn (z, τ ), Nn (z, τ ) = eiµτ Nn (z, τ ),

¯ n (z, τ ) = eiµτ M ¯ n (z, τ ) M −iµτ ¯n (z, τ ) = e ¯ n (z, τ ) N N

to be solutions of the time-dependence equation (5.4). These τ -dependent functions satisfy the relations ¯n (z, τ )a(z, τ ) Mn (z, τ ) = z −2n e−2iµτ Nn (z, τ )b(z, τ ) + N ¯ n (z, τ ) = z 2n e2iµτ N¯n (z, τ )b(z, ¯ τ ) + Nn (z, τ )¯ M a(z, τ )

(5.269) (5.270)

which are obtained from the eqs. (5.86)–(5.87). Like in the scalar case, this yields for the time evolution of the scattering data b(z, τ ) = e2iµτ b(z, 0) a(z, τ ) = ¯ ¯ a(z, 0)

a(z, τ ) = a(z, 0) ¯ τ ) = e−2iµτ b(z, ¯ 0). b(z,

(5.271) (5.272)

The evolution of the reflection coefficients is thus given by ρ(z, τ ) = e2iµτ ρ(z, 0) ρ(z, ¯ τ ) = e−2iµτ ρ(z, ¯ 0).

(5.273) (5.274)

From (5.271)–(5.272) it is clear that also for matrix DNLS the eigenvalues (i.e. the zeros of det a(z) and det ¯ a(z)) are constant as the solution evolves. Not only the number of eigenvalues, 121

but also their locations are fixed. Thus, the eigenvalues are time-independent discrete states of the evolution. The evolution of the norming constants follows from the definitions (5.111), (5.113) and the relations (5.271)–(5.272), i.e. ¯ j (τ ) = e−2iµj τ C ¯ j (0) C

Cj (τ ) = e2iµj τ Cj (0),

(5.275)

where µj =


2 1 zj − zj−1 , 2

µ ¯j =


2 1 z¯j − z¯j−1 . 2

(5.276)

Note that the expressions (5.271)–(5.272) for the evolution of the scattering data are valid only for a time-dependence (5.4) with gauge A, B = 0. If A 6= 0 and/or B 6= 0, one first determines ˆ n and R ˆ n with gauge A, B = 0 such that the evolution of the potentials Q ˆ n (τi ) = Qn (τi ) , Q

ˆ n (τi ) = Rn (τi ) R

and then uses the transformations ˆ n (τ ) e−i(τ −τi )B Qn (τ ) = e−i(τ −τi )A Q ˆ n (τ ) ei(τ −τi )A Rn (τ ) = ei(τ −τi )B R

(5.277) (5.278)

to obtain the solution of (5.1)–(5.2). For the two-component system, the 2 × 2 matrix Qn is given by (5.8) and A, B are given by (5.18). Therefore the transformations (5.277)–(5.278) give ˆ n (τ ) e−i(τ −τi )B Qn (τ ) = e−i(τ −τi )A Q =

ˆ (1) Q n n −4i(τ −τi )

(−1) e

(5.279)

ˆ (2) Q n n+1 −4i(τ −τi )

ˆ n(2) (−1) R

e

ˆ n(1) R

!

,

that is, the transformation leaves the first row of the potential invariant.

5.3 5.3.1

Vector Solitons One soliton solutions

The one soliton solution of the two-component system (5.1)–(5.2) with the symmetry conditions (5.115) and (5.116) is the reflectionless potential associated with a single octet of eigenvalues/norming constants (5.218) and the condition z¯1 = 1/z1∗ . We computed the explicit expression of these potentials for the special choice of the norming constants corresponding to γ1 · δ1 = 0. The evolution of the norming constants is determined by (5.275). Therefore, taking into account (5.279), we get from (5.245) ! (1) Qn (τ ) (5.280) (2) Qn (τ ) i sinh 2a1 sech (2(n + 1)a1 − vτ + d(0)) h iπ+2i(n+1)b1 −2iω− τ ∗ e γ 1 (0) + e−2i(n+1)b1 −inπ+2iω+ τ δ 1 (0) = 1/2  2 2 kγ 1 (0)k + kδ 1 (0)k

where

v = − sinh (2a1 ) sin (2b1 ) ,

ω± = cosh (2a1 ) cos (2b1 ) ± 1 122

(5.281)

d(0) = log sinh 2a1 −

  1 2 2 log kγ 1 (0)k + kδ 1 (0)k . 2

(5.282)

Note that this is a special type of “composit” soliton in which the two terms have different carrier frequencies but the peaks are coincident. By letting either γ 1 = 0 or δ 1 = 0 in (5.280), one obtains the solutions we referred to as “fundamental” solitons.

5.3.2

Transmission coefficients for the pure 1-soliton potential

We can use the representations (5.96) and (5.98) to reconstruct the transmission coefficients c(z) and ¯ c(z). We are interested in the reflectionless case with J = 1 when Symmetry 5.3 holds for the case Qn = −RH n . For definiteness, we consider the case when δ 1 = 0. Taking into account the symmetry (5.116), eq. (5.212) gives H H   −2(n+1) 0(up) n 2n ˆ H ¯ 0(up) ¯ Rn = −QH = 2z C + 2(−1) z ¯ C N (¯ z ) N (i/z ) 1 1 1 n 1 1 1 n+1 n+1

where we also used the condition z¯1 = 1/z1∗ following from Symmetry 5.3. Moreover, eq. (5.186) with J = J¯ = 1 and ρ(w) = 0 gives N0(up) (z) = N(up) (z) = 2 n n

−2n z¯12n z ¯ 0(up) ¯ 1 + 2(−1)n z1 z N ¯ 0(up) ˆ1 N (¯ z ) C (i/z1 )C 1 n n 2 z 2 − z¯1 z 2 + z1−2

so that, substituting into (5.96), yields c(z) = IM −

+∞ X

k=−∞

"

 H   H  0(up) −2(k+1) k 2k ˆ H ¯ 0(up) ¯ × + 2(−1) z1 C1 Nk+1 (i/z1 ) z1 ) z 2z1 C1 Nk+1 (¯

z¯2k z ¯ 0(up) ¯1 2 21 2N (¯ z1 )C z − z¯1 k

# z1−2k z ¯ 0(up) ˆ1 . + 2(−1) 2 N (i/z1 )C z + z1−2 k k

Writing the norming constants as in (5.221)–(5.222) with the constraint δ1 = 0 and using the corresponding explicit expressions (5.227)–(5.228) for the Jost functions, we obtain ! +∞ (1) (1) (2) X z −2(k+1) z¯2(k+1) z2 |γ1 |2 γ1 ¯ γ1 1 1 c(z) = IM − 4 2 (1) (2) (2) z − z¯12 ¯ γ1 γ1 |γ1 |2 k=−∞ (1 + gk )(1 + gk+1 ) ! +∞ (2) (1) (2) X z −2(k+1) z¯2(k+1) z2 |γ1 |2 −γ1 ¯ γ1 1 1 −4 2 (1) (2) (1) (1 + gk )(1 + gk+1 ) z + z1−2 −¯ γ1 γ1 |γ1 |2 k=−∞

where gn is given by (5.229) and we have taken into account that, when Symmetry 5.3 holds, gk∗ = gk . Then, using the identity   z¯12 − z12 1 1 1 = − (5.283) 2 2(k+1) −2k 1 + g (1 + gk )(1 + gk+1 ) 1 + gk+1 k 4 kγ1 k z¯1 z1 we can write ! +∞ 
 (1) (1) (2) X z 2 z¯12 − z12 z1−2 1 1 |γ1 |2 γ1 ¯ γ1 c(z) = IM − − (1) (2) (2) k|γ1 k (z 2 − z¯12 ) 1 + gk+1 γ1 γ1 |γ1 |2 k=−∞ 1 + gk ¯ ! +∞ 
 (2) (1) (2) X z 2 z¯12 − z12 z1−2 1 1 |γ1 |2 −γ1 ¯ γ1
− − . (1) (2) (1) 1 + gk 1 + gk+1 kγ1 k z 2 + z1−2 −¯ γ1 γ1 |γ1 |2 k=−∞ 123

The series in the right-hand side is convergent since lim

k→+∞

1 = 1, 1 + gk

lim

k→−∞

1 =0 1 + gk

(5.284)

and therefore we can explicilty compute the sum +∞  X

k=−∞

   1 1 1 1 − − = lim = −1 j→+∞ 1 + g−j 1 + gk 1 + gk+1 1 + gj+1

so that

c(z) = IM +

z

2

z¯12


2

− z1 kγ1 k

z1−2

  

(1) 2 γ1

+

z 2 −¯ z12

(1) (2)

γ1 γ1 ¯

(2) 2 γ1

z1−2 +¯ z12 2 )(z 2 +z −2 ) 2 −¯ (z z 1 1 (1) 2 (2) 2 γ1 γ1

(1) (2)

γ1 ¯ γ1

z 2 +z1−2

z1−2 +¯ z12 (z 2 −¯ z12 )(z 2 +z1−2 )

z 2 +z1−2

+

z 2 −¯ z12



 .

(5.285)

Note that, as expected, c(z) is analytic outside the unit circle and when z → ∞ c(z) ∼ z¯12 z1−2 IM

(5.286)

and comparing (5.286) with the asymptotic expansion of c(z) given by (5.101) we get the relation 1+

+∞ X

αk

+∞ Y

−1

(1 − αj )

= z¯12 z1−2 .

(5.287)

j=k

k=−∞

Moreover, one can easily check that det c(z) = z¯14 z1−4


z 2 + z¯1−2
(z 2 − z¯12 ) z 2 + z1−2 z 2 − z12




which means that indeed det c (z) has zeros at the proper points. Then the symmetry relation (5.135), i.e. a(z) = ¯

+∞ Y

(1 − αj )c(1/z ∗ )H

j=−∞

we get the expression for ¯ a(z)

¯ a(z) =

+∞ Y

j=−∞



z  (1 − αj ) IM +

−2


2 −2 2 z1 − z¯1 z¯1   2 kγ1 k

(1) 2 γ1

z −2 −z1−2 (1) (2)

γ1 γ1 ¯

+

(2) 2 γ1

z −2 +¯ z12

z¯12 +z1−2 (z −2 −z1−2 )(z −2 +¯ z12 )

From (5.135) it also follows det ¯ a(1/z ∗ ) =

+∞ Y

(1 − αj )2 [det c(z)]∗

j=−∞

124

z¯12 +z1−2 −2 −z −2 )(z −2 +¯ (z z12 ) 1 (1) 2 (2) 2 γ1 γ1

(1) (2)

γ1 ¯ γ1

z −2 +¯ z12

+

z −2 −z1−2

(5.288)



  .

and evaluating this relation for |z| → ∞ and taking into account that ¯ a(1/z ∗) → IM and c(z) → 2 −2 z¯1 z1 IM yields +∞ Y

(1 − αj )2 = z14 z¯1−4

(5.289)

j=−∞

so that, substituting it into (5.288), we finally get    z 2 (¯ z 2 +z −2 ) (1) 2 z¯12 (1) (2) (2) 2 z1−2 γ1 (z2 −z2 1)(z21+¯z−2 ) 1 − |z1 |4  −|γ1 | z2 −z12 + |γ1 | z2 +¯z1−2 −γ1 ¯ 4 1 1  . a(z) = |z1 | IM + ¯ 2 z1−2 (¯ z 2 +z −2 ) (1) (2) 2 z¯12 (1) (2) |γ 1 | |γ1 |2 z2 +¯ −¯ γ1 γ1 (z2 −z12 )(z12 +¯z−2 ) −2 + |γ1 | z 2 −z 2 z 1

1

1

1

(5.290)

In order to reconstruct ¯ c(z) from (5.98)and, consequently, a(z) by means of (5.136), we need ¯ 0(dn) to solve the system (5.191)–(5.196) with J = J¯ = 1 and ρ = 0, ρ¯ = 0 to determine N (z). A n few calculations yield  z−2n (1)  2n (2) n z¯1 1 γ (−1) γ ¯ 2 −2 2 1 2z −z1 1 z 2 +¯ z1 ¯ 0(dn)  z −2n . (5.291) N (z) = 2n n z1 (1) (2) n+1 z¯1 1 + z¯12 z1−2 gn γ (−1) 2γ −2 ¯ 2 z −z1

1

1

z 2 +¯ z1

Evaluating (5.230) as z → 0 and comparing it with the asymptotic expansion (5.152), we get −1 Ω−1 n = ∆n =

1 + z¯12 z1−2 gn IN 1 + gn

(5.292)

where the first equality follows from (5.203). Note that when n → −∞ (5.292) yields +∞ Y

¯12 z1−2 (1 − αj )−1 = lim ∆−1 n = z n→−∞

j=−∞

which is in agreement with (5.289). From (5.152) and (5.291)–(5.292) it follows  z−2n (1)  2n (2) n z¯1 1 γ ¯ γ (−1) −2 2 2 1 −z1 1 z 2 +¯ z1 ¯ 0(dn) (z) = 2z  z −2n ¯ (dn)  N (z) = ∆−1 2n n Nn n z1 (2) (1) n+1 z¯1 1 + gn (−1) γ 2γ −2 ¯ 2 z −z1

1

z 2 +¯ z1

1

and the potential matrix Qn is given by (5.231)–(5.232) taking into account (5.8), i.e. ! 2(n+1) (1) 2(n+1) (2) 2 z¯1 γ1 ¯ z¯1 γ1 ¯ Qn = − . 1 + gn+1 (−1)n+1 z1−2(n+1) γ1(2) (−1)n+1 z1−2(n+1) γ1(1) Substituting these expressions into (5.98) yields  −2k 2(k+1)  z1 z¯1 2 +∞ X kγ k 0 2 1 2 1  z −z1 . ¯ c(z) = IN + 4 −2(k+1) 2k z¯ z (1 + gk )(1 + gk+1 ) 0 − 1 z2 +¯z−21 kγ1 k2 k=−∞ 1

Using once more (5.283)–(5.284) we get



¯ c(z) = 

z 2 −¯ z12 z 2 −z12

0

125

0 z 2 +z1−2 z 2 +¯ z1−2

 

(5.293)

with lim ¯ c(z) = z¯12 z1−2 I

z→0

(z 2 − z¯12 )(z 2 + z1−2 ) . (z 2 − z12 )(z 2 + z¯1−2 )

det ¯ c(z) =

Finally, using the symmetry (5.136), (5.293) provides the expression for a(z)   2 2 z −z1 0 2 2 z −¯ z1 . a(z) =  z 2 +¯ z −2 0 z2 +z1−2

(5.294)

1

We conclude that the transmission coefficients for the pure 1-soliton solution of the system (5.1)–(5.2) under Symmetry 5.3 and corresponding to the set of eigenvalues/norming constants (5.233) with δ 1 = 0 in (5.221)–(5.222) (hence, according to our definitions, we are considering a fundamental soliton) are given by    (1) (2) z¯12 |γ 1 |2 z 2 (¯ z12 +z1−2 ) z1−2 |γ 1 |2 (1) (2) 4 − −γ γ ¯ + 2 −2 −2 1 1 (z 2 −z 2 )(z 2 +¯ z 2 −z1 1 − |z1 |  z 2 +¯ z1 z1 )    1 a(z) = |z1 |4 IM + ¯  . (5.295) −2 (1) 2 (2) 2 2  −2 2 2 z γ 1 1 (1) (2) (¯ z +z ) z¯1 |γ 1 | kγ1 k −¯ γ 1 γ 1 (z2 −z12 )(z12 +¯z−2 ) + 2 −2 2 z −z z 2 +¯ z 1



a(z) = 

z 2 −z12 z 2 −¯ z12

0

1

0 z 2 +¯ z1−2 2 z +z1−2

1



.

1

(5.296)

Note that, as expected, they are time-independent. Moreover, a is diagonal and does not depend on the norming constants, while ¯ a does. The role of a and ¯ a is interchanged if one takes the norming constant of the fundamental soliton to be of the form  (1)  (2) γ1 γ1 C1 = . (5.297) 0 0

5.3.3

Vector soliton interactions

The problem of a multisoliton collision can be investigated by looking at the asymptotic states as τ → ±∞. The solutions of discrete VNLS are more complex that the solutions of the scalar IDNLS because, in the vector equations, there are more degrees of freedom. In particular, vector solitons are characterized, in part, by a polarization. In direct analogy with VNLS, the system (1.9)–(1.10) reduces to the scalar IDNLS under the reduction qn = qn p

(5.298)

where p is an N -component vector with kpk = 1 which is referred to as the polarization of the reduction solution (5.298). Moreover, if qn satisfies the symmetric system, then so does Uqn where U is any unitary matrix, hence the polarization of a particular reduction solution depends on the choice of the basis. For a generic multi-soliton solution (in which the solitons travel with different velocities) we can define independently a polarization for each soliton in the long-time limit where the solitons are well-separated. Then, when τ → ±∞ qn ∼

J X j=1

126

q± j,n

where ± ± q± j,n = qj,n pj

(5.299)

± and qj,n is a one soliton solution of IDNLS. We then identify p± j as the polarization of the j-th soliton before (−) and after (+) the soliton interaction. For the matrix system (5.1)–(5.2) with Symmetry 5.3, we will write

Qn ∼

J X

Qn(j),± .

(5.300)

j=1

We can investigate the problem of a multisoliton collision proceeding in a similar way as for the continuous VNLS equation (cf. Chap. 4). We consider the solution of discrete matrix NLS (5.1)– (5.2) under Symmetry 5.3 corresponding to the scattering data {zj : |zj | > 1, Cj }Jj=1 and for simplicity we restrict ourselves to the case of fundamental solitons when all the norming constants Cj have the form ! (1) γj (0) 0 j = 1, . . . , J. (5.301) Cj (0) = (2) γj (0) 0 Let us fix the values of the soliton parameters for τ → −∞, i.e. for each eigenvalue zj we assign (j),− a matrix S− . The matrices S− j that completely determines Qn j are the “effective” norming (j),−

constants for the pure one-soliton potential Qn . For τ → +∞ we denote the corresponding matrices by S+ j . For definiteness, let us assume that the discrete eigenvalues are such that the velocities of the individual solitons satisfy the condition v1 < v2 < · · · < vJ (cf. (5.281)). Then, as τ → −∞ the solitons are distributed along the n-axis in the order corrsponding to vJ , vJ−1 , . . . , v1 ; the order of the soliton sequence is reversed as τ → +∞. In order to determine the result of the − interaction between solitons, i.e. to calculate S+ j given Sj , we trace the passage of the Jost functions through the asymptotic states. We denote the soliton coordinates at the instant of time τ by nj (τ ) (|τ | is assumed large enough so that one can talk about individual solitons). If τ → −∞ then nJ  nJ−1  · · ·  n1 . The matrix-valued function φn (zj ) has the form   IN φn (zj ) ∼ zjn n  nJ . 0 After passing through the J-th soliton it will be of the form   IN n aJ (zj , S− φn (zj ) ∼ zj J) 0 where aJ (z, S− J ) is the transmission coefficient relative to the J-th soliton and the additional argument is to take into account the dependence on the relative norming constant. Repeating the argument in tracing the passage of the Jost functions through the solitons J − 1, J − 2, . . . , j + 1, we find that   IN − − n φn (zj ) ∼ zj aj+1 (zj , S− j+1 )aj+2 (zj , Sj+2 ) . . . aJ (zj , SJ ) 0   Y J IN n ∼ zj al (zj , S− nj+1  n  nj l ) 0 l=j+1 right

127

where the notation right indicates that the product is performed such that the matrix with index ` occurrs to the right of the matrix with index ` − 1. Note that due to our choice of the norming constants, according to (5.296) the transmission coefficients a` (z, S− ` ) are diagonal and independent of S− ` . Since j-th soliton corresponds to a bound state, passing through the j-th soliton yields     0 IN −n n S− zj → zj j IM 0 and therefore φn (zj ) ∼ zj−n



0 IM



S− j

J Y

al (zj , S− l )

nj  n  nj−1 .

(5.302)

l=j+1 right

On the other hand, starting from n  n1 and proceeding in a similar way we find for the Jost function ψ the following asymptotic behavior   0 − − cj−1 (zj , S− (5.303) ψ n (zj ) ∼ zj−n j−1 )cj−2 (zj , Sj−2 ) . . . c1 (zj , S1 ) IM  j−1  0 Y cl (zj , S− nj  n  nj−1 (5.304) ∼ zj−n l ) IM l=1 left

where we have used (5.88) and the notation left indicates that the product is performed such that the matrix with index ` occurrs to the left of the matrix with index ` − 1. Assuming that at an eigenvalue φn (zj ) = ψ n (zj )Bj and comparing (5.302) and (5.304), we get j−1 Y

Bj ∼

J
−1 − Y cl (zj , S− al (zj , S− ) S j l ) l

l=1 right

τ → −∞.

(5.305)

l=j+1 right

Proceeding in a similar fashion as τ → +∞ and taking into account that the order of solitons is reversed, we get Bj ∼

J Y

l=j+1 left

Y
−1 + j−1 al (zj , S+ cl (zj , S+ ) Sj l ) l

τ → +∞

(5.306)

l=1 left

and the comparison between the two representations (5.305)–(5.306) for Bj yields S+ J =

J−1 Y

l=1 right

Y
−1 − J−1 −1 ) cl (zJ , S− SJ (al (zJ , S+ l l ))

(5.307)

l=1 right

and for j = 1, . . . , J − 1 S+ j =

J Y

l=j+1 right

cl (zj , S+ l )

j−1 Y

l=1 right

j−1 J Y
−1
−1 Y − . al (zj , S+ ) ) a (z , S cl (zj , S− l j l ) l l l=j+1 right

(5.308)

l=1 right

Since al actually do not depend on S the formulae (5.307) and (5.308) are effective to determine + − − S+ j in terms of Sl . Indeed, given Sj for j = 1, . . . , J, (5.307) allows us to determine SJ and in 128

+ (5.308) only the knowledge of S+ l for l > j is required in order to determine Sj . Hence one can + iteratively solve the system (5.307)–(5.308) for Sj , j = 1, . . . J. Let us consider in details the interaction of two solitons. In this case (5.307)–(5.308) give − −1 − S+ S2 a1 (z2 )−1 2 = c1 (z2 , S1 )

(5.309)

S+ 1

(5.310)

=

c2 (z1 , S+ 2)

S− 1

a2 (z1 )

where we have taken into account that, due to (5.294), the matrices aj (z) are independent of Sj . Note that the formulae (5.309)–(5.310) are not symmetric with respect to the exchange of the subscripts 1 and 2. However, such a notation expresses the invariance of the system under the substitution τ → −τ, Qn → QH n (here a “fast” soliton becomes a “slow” soliton and vice-versa). Taking into account the explicit expressions of aj (z) and cj (z, Sj ) for the pure soliton case, one can solve (5.309) for S+ 2 and then substitute it into the right-hand side of (5.310) in order to get S+ . We observe that the elements in the first column of matrix Sj are proportional to the vectors 1 γ j , which, in their turn, are related to the polarization of the j-th vector soliton (cf. (5.280)). ± Therefore if we denote by s± j the vectors composing the first column of the matrices Sj , the unit polarization vectors before and after the interaction are written, respectively, as
∗
∗ s+ s− j j 0 pj = + (5.311) pj = −

s

s j

j

Taking into account the explicit expression of a1 (z2 ) and a2 (z1 ) provided by (5.296), from (5.309)– (5.310) it follows
−1 − z22 − z¯12 c1 (z2 , S− s2 1) 2 2 z2 − z1 z12 − z22 − s+ c2 (z1 , S+ 1 = 2 2 )s1 . z1 − z¯22 s+ 2 =

(5.312) (5.313)

Introduce + 2 2 s z2 − z¯12 2 H −1
2 p c (z2 , S− ) H c−1 (z2 S− )p2 . χ = 2 = 2 2 1 1 1 1 2 − z2 − z1 s 2

(5.314)

2

From (5.285) it follows − c−1 1 (z2 , S1 ) =

(z 2 z14 z¯1−4 22 (z2

+

z22

− −

1 + z1−2

z¯12 )(z22 z12 )(z22

+ +

z1−2 ) z¯1−2 )

 (1) 2 p 1   (1)∗ (2) p1 p1

    

IM + z22 z1−2 

 (1) (2)∗ p1 p1  (2) 2   p1


 z¯12 − z12 

  (2) 2 (1) (2)∗ −p p p 1 1 1 1    + z22 − z¯12 −p(1)∗ p(2) p(1) 2 1 1 1

and therefore one can explicitly express χ2 in terms of the parameters of the colliding solitons 2 2 o (z − z¯22 )2 (¯ z12 + z¯2−2 ) n 2 2 ∗ χ2 = z1−2 z¯12 12 1 + α |p · p | + β |W (p p )| 2 1, 2 1 −2 2 2 (¯ z1 − z¯2 )2 (z1 + z¯2 ) (¯ z 2 − z 2 )(z 2 − z¯22 ) (¯ z12 − z12 )(¯ z22 − z22 ) α = z2−2 z¯2−2 2 1 −21 22 , β = (z12 − z¯22 )(¯ z12 − z22 ) (z1 + z2 )(¯ z1 + z¯2−2 ) 129

(5.315) (5.316)

where, as usual, (1) (2)

(2) (1)

W (p1 , p2 ) = p1 p2 − p1 p2 . Moreover, from (5.311) and (5.314) it follows p02 = so that we finally get p02

1 (z12 − z¯22 )2 (¯ z12 + z¯2−2 ) = χ (¯ z12 − z¯22 )2 (z12 + z¯2−2 )


∗ 1 −2 2 z12 − z¯22 −1 c1 (z2 , S− p2 z1 z¯1 2 1) 2 χ z¯1 − z¯2

  (z12 − z¯12 ) ¯12 − z12 −2 −2 z ∗ (p1 · p2 ) p1 + 2 p2 + z1 z¯2 2 W (p1 , p2 ) p ¯1 (z1 − z¯22 ) z¯1 + z¯2−2 (5.317)

where we introduced the following notation: given a vector c = c(1) , c(2) as  (2) ∗ c ¯ c= −c(1)


T

, the vector ¯ c is defined

(5.318)

Analogously, from (5.313) and (5.311) it follows that
∗
s+ (¯ z22 − z¯12 ) s− 1 1 + −2 2 ∗ 0 p1 = + = z2 z¯2 + c2 z1 , S2 p1 2 2 (z2 − z¯1 ) s1 s1

and since


 z¯22 − z22 

 ∗  0(1) 2 0(2) 0(1) p p p 1 2 2 2   2 −2   + c2 (z1 , S+  2 ) = IM + z1 z2 0(2) 2  z12 − z¯22 p0(1) p0(2) ∗ p 2 2 2   2  ∗ 0(2) 0(1) 0(2) − p2 p2   p2 1  + 2  ∗   0(1) 2 z1 + z2−2 −p0(1) p0(2) p2 2 2 

one can check that

+ 2 2 s z1 − z22 2 T
1 p c2 (z1 , S+ ) H c2 (z1 , S+ )p∗ = 1 − 2 = 2 1 1 2 2 2 z1 − z¯2 χ2 s 1

and p01 = z22 z¯2−2

  2 (¯ z22 − z¯12 ) z¯22 − z22 ¯22 ) −2 −2 (z2 − z 0∗ 0 0∗ 0 p χ p + (p · p ) p − z ¯ z W (p , p ) 1 1 2 2 1 2 (z22 − z¯12 ) z22 − z¯12 1 2 2 (¯ z1−2 + z¯22 )

which, taking into account (5.317), gives the expression of p01 in terms of p1 and p2 . Without loss of generality, one can parametrize the incoming polarization vectors of the solitons as follows     ρeiϕ 1 p p1 = , p2 = (5.319) 0 1 − ρ2 130

with 0 ≤ ρ ≤ 1 and 0 ≤ ϕ ≤ 2π. Hence from (5.315)–(5.316) and (5.317) it follows that the polarization vectors after the interaction are such that 2 1 4 −4 (z12 − z¯22 )2 2 0(1) 2 ρ p2 = 2 z¯1 z1 2 χ (¯ z1 − z¯22 )2 2 z12 + z¯2−2 ) 1 (z12 − z¯22 )(¯ 0(2) 2 (1 − ρ2 ) = p2 χ2 (¯ z12 − z¯22 )(z12 + z¯2−2 )

with χ=

z1−1 z¯1

2 (z1 − z¯22 )2 (¯
z12 + z¯2−2 )  2 2 −2 1 + αρ + β 1 − ρ (¯ 2 2 2 2 z1 − z¯2 ) (z1 + z¯2 )

Note that, like in the continuous case, the magnitudes of the soliton polarizations do not change only in the case when their initial polarizations are either parallel or orthogonal. In the continuous limit, i.e. for zj = e−ikj h as h → 0, the formula (5.317) coincides with (4.140). In Fig. 5.1 we 0(1) 0(1) plot both |p1 |2 and |p2 |2 as functions of ρ2 for some specific choice of the eigenvalues. The continuous curve represents the continuous limit. 2 Èp’H1L 2 È

Èp’1H1LÈ2 1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.2

0.4

0.6

0.8

1

Ρ2

0.2

0.4

0.6

0.8

1

Ρ2

Figure 5.1: Intensity shift induced by two-soliton interaction of 2-component MNLS. The soliton parameters are: zj = exp (aj + ibj )h with a1 = 1, a2 = 2, b1 = −b2 = 0.1 and h = 1, .5, .1. The continuous curves represents the continuous limit, i.e. h → 0. The initial polarizations are given by (refinitiald).

5.4

Conserved Quantities

In this section we calculate polynomial (in Qn and Rn ) conserved quantities of the system (5.1)– (5.2). The method given here applies to the general matrix system (5.1)–(5.2) also for M 6= N . We proved that in the gauge A, B = 0 the scattering coefficient a(z) is time-independent. Since a(z) is analytic for |z| > 1, it admits a Laurent expansion whose coefficients are constants of the motion, as well. From the representation (5.92) for a(z), it follows that the quantities Γj =

+∞ X

Qn M(dn),−2j+1 n

n=−∞

131

(5.320)

(dn),−2j+1

(dn)

where Mn are the coefficients of the Laurent expansion of Mn (z), are conserved for any integer j ≥ 1. Hence, in order to find conserved quantities, we need only to express such coefficients in terms of the potentials, which are given by (5.64)–(5.65). For instance, the first two coefficients are Mn(dn),−1

Mn(dn),−3

= Rn−1 ,

= Rn−2 + Rn−1

n−2 X

Qk Rk−1

k=−∞

and therefore the first two (matrix) constants of the motion are given by Γ1 =

+∞ X

Qn Rn−1 ,

Γ2 =

n=−∞

+∞ X

Qn Rn−2 +

n=−∞

+∞ X

Qn Rn−1

n=−∞

n−2 X

Qk Rk−1 .

(5.321)

k=−∞

The scattering coefficient ¯ a(z) is also a constant of the motion in the gauge A, B = 0 and proceeding exactly as before one can obtain a second set of conserved quantities given by ¯= Γ

+∞ X

¯ (up),2j−1 Rn M n

n=−∞

for any j ≥ 1. Hence, from (5.67)–(5.68) it follows ¯1 = Γ

+∞ X

Rn Qn−1 ,

¯2 = Γ

+∞ X

Rn Qn−2 +

Rn Qn−1

n=−∞

n=−∞

n=−∞

+∞ X

n−2 X

Rk Qk−1 .

(5.322)

k=−∞

Note also that, taking into account the τ -dependence of the scattering matrix defined by (5.85), the determinant of the scattering matrix is a constant of the motion in the gauge A, B = 0, i.e.     ¯ 0) ¯ τ) a(z, 0) b(z, a(z, τ ) b(z, . = det det b(z, 0) ¯ a(z, 0) b(z, τ ) ¯ a(z, τ ) To obtain a conserved quantity in terms of the potentials, we observe that from (5.81)–(5.83) it follows   +∞ Y ¯ ab (5.323) det (IM − Rj Qj ) = det b¯ a j=−∞

and therefore (5.323) is a constant of the motion when A, B = 0. In the general gauge A, B 6= 0, the method illustrated so far generates conserved quantities in ˆ n and R ˆ n (cf. (5.5)) and the gauge transformation allows to terms of the transformed potentials Q express such constants of the motions in terms of the original potentials Qn and Rn . In particular, the system (1.9)–(1.10) is a reduction of the system (5.1)-(5.2) with the rescaling (5.19) and A and B given by (5.18). For instance, under the reduction (5.6)–(5.8), the trace of (5.321) is 2N −1

+∞ X N h i X (j) (j) Qn−1 Rn(j) + Q(j) n Rn−1

n=−∞ j=1

or, with the rescaling (5.19) +∞ X N X 

qn−1 · rn + qn · rn−1

n=−∞ j=1

132



(5.324)

which is therefore a constant of the motion. Similarly, by taking the trace of (5.321) one obtains the conserved quantity  +∞  X
2
i2 h2 h qn−1 · rn − qn · rn−1 + 2 (qn · rn ) qn−1 · rn−1 qn−2 · rn + qn · rn−2 − 2 n=−∞

(5.325)

and so on. ˆ nQ ˆ n ) = det(IM − Rn Qn ), it follows that it is a conserved quantity Finally, since det(IM − R regardless of the gauge and under the reduction (5.6)–(5.8) this conserved quantity can be written as Γ=

+∞ Y

j=−∞


1 − h2 qn · rn .

133

(5.326)

Appendix A

Summation by Parts Formula P+∞ P+∞ +∞ +∞ Lemma A.1 For any sequences {aj }j=−∞ and {bj }j=−∞ such that −∞ aj < ∞ and −∞ bj < ∞     ! ! n n k k−1 n n     X X X X X X aj . bj = (A.1) bk ak ak bk −     k=−∞

j=−∞

k=−∞

k=−∞

k=−∞

j=−∞

Proof A.1

   n k k−1  X  X X  bj = (aj − aj−1 ) ak bj     j=−∞ j=−∞ j=−∞ k=−∞ k=−∞       k−1 k−1 k n k   X X X X X  bj  aj   bj  −  = aj     j=−∞ j=−∞ j=−∞ j=−∞ k=−∞   n k   X X bk − aj   j=−∞ k=−∞   ! ! k n n n   X X X X aj = bk ak bk −    n  X

k−1 X

 

k=−∞

j=−∞

k=−∞

k=−∞

The formula for a finite lower bound and an infinite upper bound can be obtained by essentially the same approach. If ak = bk , Lemma A1 yields the identity   !2 n n k−1 n  X X  1 X 1 X 2 − bk bk bj = bk  2  2 j=−∞ k=−∞

k=−∞

k=−∞

therefore

 n  X

k=−∞

k−1 X

 

1 bk bj ≤   2 j=−∞

n X

k=−∞

bk

!2

.

Now, we use the summation by parts formula (A.1) to prove a generalization of (A.2). 134

(A.2)

+∞

Lemma A.2 For any real positive sequence {bj }j=−∞ such the the series and for any m ∈ N0 n X

k=−∞



bk 

k−1 X

j=−∞

Proof A.2 We show by induction that n X

k=−∞

where



bk 

k−1 X

j=−∞

bj 

bj 

−∞ bj

is convergent

m+1  n 1  X  . bj ≤ m + 1 j=−∞

m

m

P+∞

(A.3)

 m+1 n 1  X  = bj − Bn(m) m + 1 j=−∞ (m)

Bn(m) ≥ 0,

Bn+1 ≥ Bn(m) .

(0)

The result is trivially true for m = 0 with Bn = 0. Assuming it holds for m − 1 we have m  k−1 n X X bj  bk  j=−∞

k=−∞

=

n X

k=−∞





   bk

k−1 X

j=−∞

m−1  k−1  X bj  bj  j=−∞

     m−1      n k−1 n k−1   k−1  X   X X X X     bj bj − bj  bk =  bk        j=−∞

j=−∞

k=−∞



k=−∞

j=−∞



m−1  n k−1 X  2 X   − bj bk  k=−∞

j−1 X

l=−∞

!m−1    bl 

j=−∞

and using the inductive hypothesis we get that the result holds also for m with   m−1  n k−1   X  X m  (m−1) bk + b2k  bj  Bn(m) =  Bn(m−1) − Bk−1 . m+1 j=−∞ k=−∞

Note that for bj ≥ 0, the estimate n X

k=−∞



bk 

k−1 X

j=−∞

m

bj 



≤

n X

j=−∞

m+1

bj 

is elementary. However, with estimate we would only be able to estabilish that the Neumann series −1 for the Jost functions converge for kQk1 , kRk1 < 1. It is the additional factor (m + 1) in (A.3) which extends the convergence of these series to potentials Q, R ∈ L1 .

135

Appendix B

Transmission of the Jost function through a localized potential ˜ for VNLS Suppose it is given the asymptotic form of the Jost function M   IN ˜ M(x, k) ∼ aL (k) x  −L 0

(B.1)

˜ is and we want to find the “transmitted” function through a barrier of finite extension 2L. M solution of the integral equation (4.16)   Z +∞ IN ˜ ˜ G+ (x − ξ, k)Q(ξ)M(ξ, k)dξ (B.2) M(x, k)= aL (k) + 0 −∞ where G+ (x, k) is given by (4.15), i.e.    Z x  ˜ (dn) (ξ, k) IN q(ξ)M ˜ M(x, k)= aL (k) + ˜ (up) (ξ, k) dξ. 0 e2ik(x−ξ) r(ξ)M −∞

(B.3)

ˆ Introduce M(x, k) such that ˜ ˆ M(x, k) = M(x, k)aL (k).

(B.4)

ˆ solves the integral equation M ˆ M(x, k)=



IN 0



+

Z

x

−∞



ˆ (dn) (ξ, k) q(ξ)M 2ik(x−ξ) ˆ (dup) (ξ, k) e r(ξ)M



dξ

and, according to (4.41), the transmssion coefficient through the barrier is given by a(k) = IN +

Z

+∞

ˆ (dn) (ξ, k)dξ. q(ξ)M

−∞

Therefore ˆ (up) (x, k)a(k) M

136

xL

(B.5)

and from (B.4) it follows ˜ (up) (x, k) ∼ a(k)aL (k). M Moreover, the integral equation (B.5) yields ˆ (dn) (x, k) → 0 M

x → +∞

for k in the UHP and we conclude that the transmitted Jost function looks like   IN ˜ a(k)aL (k) x  −L. M(x, k) ∼ 0 A similar result can be obtained for the discrete matrix NLS equation.

137

(B.6)

(B.7)

Appendix C

Scattering theory for the discrete Schr¨ odinger equation It is well-known that the stationary solutions of IDNLS equation (1.8) can be related to the solutions of a discretized version of the Schr¨odinger equation. Indeed, let us consider the differentialdifference equation i

(qn+1 + qn−1 − 2qn ) dqn + + |qn |2 (qn+1 + qn−1 ) = 0 dt h2

(C.1)

and let us look for stationary solutions of the form qn = wn ei(λ− h2 )t 2

(C.2)

for some λ ∈ R and where wn does not depend on t. Then wn satisfies the equation ˜ n (wn+1 + wn−1 ) γn = λw

(C.3)

where ˜ = h2 λ λ

(C.4) 2

2

γn = 1 + h |qn | .

(C.5)

Note that, if qn → 0 as n → ±∞, then γn → 1 as n → ±∞. Let us introduce a new function ϕn defined by wn = cn ϕn

(C.6)

with cn → 1 for n → −∞. Then ϕn satisfies γn and requiring

cn cn−1

cn+1 cn−1 ˜ n ϕn+1 + γn ϕn−1 = λϕ cn cn

(C.7)

= γn , or equivalently, cn =

n Y

j=−∞

138

γj

(C.8)

yields the discrete Schr¨ odinger equation ˜ n αn ϕn+1 + ϕn−1 = λϕ

(C.9)

αn = γn γn+1 .

(C.10)

with potential

C.1

Direct Scattering Problem

Let us write the eq. (C.9) in the form
αn ϕn+1 + ϕn−1 = z + z −1 ϕn .

(C.11)

If αn → 1 as n → ±∞ this equation is asymptotic to


ϕn+1 + ϕn−1 = z + z −1 ϕn

therefore it is natural to introduce the solutions satisfying the boundary conditions φ¯n ∼ z −n ψ¯n ∼ z n

φn ∼ z n , ψn ∼ z −n ,

n → −∞ n → +∞.

(C.12) (C.13)

¯ n (z) = z −n φ¯n (z) M ¯n (z) = z −n ψ¯n (z) N

(C.14)

It is also convenient to define the functions Mn (z) = z −n φn (z), Nn (z) = z

−n

ψn (z),

(C.15)

satisfying the following difference equation

with boundary conditions


z αn vn+1 + z −1 vn−1 = z + z −1 vn ¯ n ∼ z −2n Mn ∼ 1, M ¯n ∼ 1 Nn ∼ z −2n , N

n → −∞ n → +∞.

(C.16)

(C.17) (C.18)

Now we construct summation equations for these solutions using the method of the Green’s functions. As usual, we represent the solutions of (C.16) in the form vn = w +

+∞ X

Gn−k (z)(αk − 1)vk+1

(C.19)

k=−∞

where Gn (z) satisfies the difference equation zGn+1 + z −1 Gn−1 − (z + z −1 )Gn = −zδn,0 . Let us represent Gn and δn,0 as Fourier integral I 1 Gn = pn−1 gˆ(p)dp 2πi |p|=1 139

(C.20)

(C.21)

1 δn,0= 2πi

I

pn−1 dp.

(C.22)

|p|=1

From eq. (C.20) it then follows gˆ(p) = −

z 2 p2

z2p − (z 2 + 1)p + 1

gˆ has poles for p = 1 and p = z −2 and the value of the integral in (C.21) and, consequently, of the Green’s function Gn , depends on the location of the poles of gˆ. When |z| = 1 both poles are on the contour of integration and therefore we consider two different deformations of such contour, C out , enclosing both 1 and z −2 , as well as p = 0, and C in , enclosing p = 0 but neither 1 nor z −2 . Correspondingly, we have the two Green’s functions

z2 z2 1 − z −2n = θ(n − 1) 1 − z −2n 2 2 1−z 1−z 2
z ¯ n (z) = −θ(−n − 1) G 1 − z −2n 1 − z2

Gn (z) = θ(n)

(C.23) (C.24)

solving the difference equation (C.20). As before, θ(n) denotes the discrete version of the Heaviside function, i.e.  n X 1 n≥0 θ(n) = δk,0 = . 0 n<0 k=−∞

¯ n (z) is analytic inside and Note that Gn (z) is analytic outside the unit circle, G Gn (z) = −θ(n − 1) + 0(z −2 ) ¯ n (z) = −θ(−n − 1) z 2 + 0(z 4 ) G

|z| → ∞ z → 0.

¯n can Taking into account the asymptotics (C.17)–(C.18), the summation equations for Mn and N be written as Mn (z) = 1 +

n−1   X z2 (α` − 1) 1 − z −2(n−`) M`+1 (z) 2 1−z

(C.25)

`=−∞

¯n (z) = 1 − N

+∞   X z2 ¯`+1 (z). (α` − 1) 1 − z −2(n−`) N 2 1−z

(C.26)

`=n+1

¯n (z) are even functions of z. We will prove that if the potential Note that both Mn (z) and N (αn − 1) decays sufficiently rapidly as |n| → ∞, the summation equations (C.25)-(C.26) are welldefined and admit unique solution. Moreover, Mn (z) is analytic for |z| > 1 and continuous for ¯n (z) is analytic for |z| < 1 and continuous for |z| = 1. Under this assumption, Mn |z| = 1, while N has a convergent Laurent series expansion in the annulus centered on z = 0 Mn (z) = Mn(0) + z −2 Mn(−2) + . . .

(C.27)

and substituting this expansion into the summation equation (C.25) and matching the corresponding powers of z −2 yields Mn(0)

=1−

n−1 X

`=−∞

140

(0)

(α` − 1) M`+1

or, equivalently, (0)

Mn(0) = αn Mn+1 . Taking into account that Mn → 1 as n → −∞, we finally get n−1 Y

Mn(0) =

α−1 ` .

(C.28)

`=−∞

¯n (z) is analytic inside the unit circle, it admits a Taylor series expansion From the other side, since N about z = 0 ¯n (z) = N ¯ (0) + z 2 N ¯ (2) + . . . N n n

(C.29)

and substituting this expansion into the summation equation (C.26) and matching the corresponding powers of z 2 yields ¯n(0) = 1. N

C.1.1

Existence and analyticity of the Jost functions

It is convenient to introduce ˆ n (z) = αn−1 Mn (z) M

(C.30)

satisfying the summation equation ˆ n (z) = 1 + M

n−1 X z2 1 − z2

1 − α−1 `−1

`=−∞




 ˆ ` (z) 1 − z −2(n−`+1) M

(C.31)

and look for solutions in the form of a Neumann series ˆ n (z) = M

+∞ X

ˆ (j) (z) M n

(C.32)

j=0

where ˆ (0) = 1 M n ˆ (j+1) (z) = M n

n−1 X z2 2 1−z

`=−∞

1 − α−1 `−1


 ˆ (j) (z) 1 − z −2(n−`+1) M `

j ≥ 0.

(C.33)

For any z such that |z| ≥ 1 and for any positive integer k 2 2 z |z| −2k (1 − z ) ≤ 2 2 1 − z2 |z| − 1 and using this bound and the summation by parts formula that " #j " n−1 X 1 |z|2 ˆ (j) 2 2 Mn (z) ≤ j! |z| − 1

`=−∞

141

(A.3), one can show by induction on j

1 − α−1 `−1

#j

.

(C.34)

Hence, provided z 6= ±1 and the potential αn is such that C=

+∞ X 1 − α−1 < ∞

(C.35)

n

n=−∞

the Neumann series (C.32) is majorized by a uniformly convergent series and is itself uniformly convergent with the bound 2 2 |z| C ˆ (C.36) Mn (z) ≤ e |z|2 −1 .

This yields existence and analyticity of the Jost function Mn (z) for |z| ≥ 1, z 6= 1. From the other side, the kernel of the summation equation (C.31) Hn (z) = satisfies


z2 1 − z −2n 2 1−z

|Hn (z)| ≤

n−1 X

−2k

|z|

n≥1

≤n

(C.37)

(C.38)

k=0

for any z outside the unit circle, i.e. |z| ≥ 1. Using this bound, one can show by induction that for any j ≥ 0 " n−1 #j 1 X ˆ (j) −1 (n − ` + 1) 1 − α`−1 . (C.39) Mn (z) ≤ j! `=−∞

Indeed, from (C.33) it follows

n−1 X ˆ (j+1) 1 − α−1 |Hn−`+1 (z)| M ˆ (j) (z) (z) ≤ M n `−1 ` `=−∞

and using the bound (C.38) and the inductive hypothesis yields #j " `−1 n−1 X 1 X ˆ (j+1) −1 −1 (` − k + 1) 1 − αk−1 (n − ` + 1) 1 − α`−1 (z) ≤ M n j! k=−∞ `=−∞ " `−1 #j n−1 X X 1 . (n − ` + 1) 1 − α−1 ≤ (n − k + 1) 1 − α−1 `−1 k−1 j! `=−∞

k=−∞

The summation by parts formula (A.3) completes the induction. We conclude that ˆ Mn (z) ≤ ePn

(C.40)

where

Pn =

n−2 X

`=−∞

+∞ X ≤ . (n − `) 1 − α−1 |n − `| 1 − α−1 ` `

(C.41)

`=−∞

However, Pn diverges as n → ±∞ and therefore the convergence of the Neumann series is ensured for any finite n provided the potential satisfies the condition +∞ X

n=−∞

< +∞. (1 + |n|) 1 − α−1 n 142

(C.42)

C.1.2

Scattering Data

Let us define the Wronskian of two functions un and vn as W (un , vn ) = un+1 vn − un vn+1 .

(C.43)

If ϕn and ϕ˜n are any two solutions of the scattering problem (C.11), their Wronskian satisfies the following recursion relation W (ϕn+1 , ϕ˜n+1 ) =

1 W (ϕn , ϕ˜n ) . αn+1

Hence, for any non-negative integer j we have # " j Y

−1 αn−s W φn−j−1 (z), φ¯n−j−1 (z) W φn (z), φ¯n (z) = s=0

and in the limit j → +∞, taking into account the asymptotics (C.12), we get n

Y W φn (z), φ¯n (z) = z − z −1 α−1 s .

(C.44)

s=−∞

Analogously, from the asymptotics (C.13) it follows W (ψ¯n (z), ψn (z)) = z − z −1

Y
+∞

αs .

(C.45)

s=n+1

The relations (C.44)–(C.45) prove that φn and φ¯n are linearly independent, as are ψn and ψ¯n . Since the scattering problem (C.11) is a linear second order difference equation, the last two can be written as linear combination of the former ones (or vice-versa), i.e. φn (z) = b(z)ψn (z) + a(z)ψ¯n (z) φ¯n (z) = a ¯(z)ψn (z) + ¯b(z)ψ¯n (z)

(C.46) (C.47)

for any z such that all four eigenfunctions exist. In particular, these relations hold for |z| = 1 and define the scattering coefficients a, a ¯, b and ¯b. In terms of the Jost functions, eqs. (C.46)–(C.47) are written as ¯n (z)a(z) + Nn (z)b(z) Mn (z) = N ¯ n (z) = Nn (z)¯ ¯n (z)¯b(z) M a(z) + N

(C.48) (C.49)

¯n (z) + Nn (z)ρ(z) µn (z) = N ¯n (z)¯ µ ¯n (z) = Nn (z) + N ρ(z)

(C.50)

or also

(C.51)

where µn (z) =

Mn (z) , a(z)

µ ¯n (z) =

143

¯ n (z) M a ¯(z)

(C.52)

and the reflection coefficients have been introduced b(z) a(z)

ρ(z) =

ρ¯(z) =

¯b(z) . a ¯(z)

(C.53)

The scattering coefficients can be related to the Wronskian of the Jost functions. Indeed, from (C.46)-(C.47) it follows W (φn (z), ψn (z)) = a(z)W (ψ¯n (z), ψn (z)) and using (C.45) we get z a(z) = 2 z −1

"

+∞ Y

α−1 s

s=n+1

#

W (φn (z), ψn (z)) .

(C.54)


W φ¯n (z), ψ¯n (z) .

(C.55)

Analogously, one can show that z a ¯(z) = 1 − z2

"

+∞ Y

α−1 s

s=n+1

#

Comparing (C.44) and (C.45) yields
W φn (z), φ¯n (z) =

"

+∞ Y

α−1 s

s=−∞

#


W ψ¯n (z), ψn (z)

and using (C.46)–(C.47) gives the following characterization relation for the scattering coefficients a(z)¯ a(z) − b(z)¯b(z) =

+∞ Y

α−1 s .

(C.56)

s=−∞

C.1.3

Symmetries

The scattering problem (C.11) is symmetric for the exchange of z to z −1 and so are the asymptotic conditions (C.12)–(C.13), therefore φ¯n (z) = φn (1/z)

ψ¯n (z) = ψn (1/z)

(C.57)

or, analogously ¯ n (z) = z −2n Mn (1/z) , M

¯n (1/z) . Nn (z) = z −2n N

(C.58)

In its turn, the symmetry between the eigenfunctions induces, due to (C.46)–(C.47), the following symmetry in the scattering data ¯b(z) = b (1/z)

a ¯(z) = a (1/z) ,

(C.59)

and consequently ρ¯(z) = ρ (1/z) .

(C.60)

In addition, when the potential αn is real, if ϕn (z) solves the scattering problem (C.11), so does ϕ∗n (1/z ∗), therefore, taking into account the asymptotic conditions (C.12)–(C.13), we get the following symmetry relations φ¯n (z) = φ∗n (1/z ∗ ) ,

ψ¯n (z) = ψn∗ (1/z ∗ ) . 144

(C.61)

or, analogously, ¯ n (z) = z −2n Mn∗ (1/z ∗ ) , M

¯n∗ (1/z ∗) Nn (z) = z −2n N

(C.62)

As a consequence, for real potential the following symmetries between the scattering data hold: ¯b(z) = b∗ (1/z ∗) ,

a ¯(z) = a∗ (1/z ∗) .

(C.63)

We can now reconstruct the scattering data by means of integral representations. Recalling the ¯n (z)a(z) can written as summation equations (C.25)–(C.26) the quantity Mn (z) − N ¯n (z)a(z) = 1 − a(z) + Mn (z) − N

+∞ X

k=−∞

+

z2 1 − z2

+∞ X

k=−∞

  ¯ n−k (z) Mk+1 (z) − N ¯k+1 (z)a(z) (αk − 1) G

  (αk − 1) 1 − z −2(n−k) Mk+1 (z)

where we used the identity ¯ n (z) = Gn (z) − G


z2 1 − z −2n . 1 − z2

¯n (z)a(z) = Nn (z)b(z), and therefore From the other side, we also have Mn (z) − N +∞   X z2 −2(n−k) (α − 1) 1 − z Mk+1 (z) Nn (z)b(z) = 1 − a(z) + k 1 − z2 k=−∞

+

+∞ X

¯ n−k (z)Nk+1 (z)b(z) (αk − 1) G

k=−∞

or, taking into account the symmetry (C.62), # " +∞ X ∗ ∗ ∗ 2(n−k−1) ∗ −2n ¯ ¯ ¯ Nn (1/z ) − z (αk − 1) Gn−k (z)Nk+1 (1/z ) b(z) z k=−∞

+∞   X z −2(n−k) (α − 1) 1 − z Mk+1 (z). k 1 − z2 2

= 1 − a(z) +

k=−∞

From the summation equation (C.26) it follows that the term in square brackets in the left-hand side is equal to 1, so that a(z) = 1 +

+∞ X z2 (αk − 1) Mk+1 (z) 1 − z2

(C.64)

k=−∞

b(z) = −

+∞ X z2 z 2k (αk − 1) Mk+1 (z) 1 − z2

(C.65)

k=−∞

and a ¯, ¯b can be reconstructed using the symmetries. Note that from (C.25) and (C.64)–(C.65) it also follows that a and b are even functions of z a(−z) = a(z),

b(−z) = b(z). 145

(C.66)

The same holds for a ¯ and ¯b. Note that, taking into account (C.28), the summation representation (C.64) yields +∞ X

a(z) = 1 −

(αk − 1)

k=−∞

k Y

−2 α−1 ). ` + O(z

(C.67)

`=−∞

The representation (C.64) proves that a(z) has the same analytic properties as Mn (z), i.e. it is analytic for |z| > 1 and continuous for |z| = 1 and a ¯(z) inherits the analytic properties from Mn (z −1 ), i.e. it is analytic for |z| < 1 and continuous for |z| = 1. Note that b(z) and ¯b(z) cannot in general be prosecuted off the unit circle.

C.1.4

Eigenvalues and norming constants

We define a proper eigenvalue for the scattering problem (C.11) to be a value of z for which the scattering problem admits bounded solutions going to zero as n → ±∞. From the equations (C.54)–(C.55), it follows that if zj , with |zj | > 1, is a zero of a(z) then W (φn (zj ), ψn (zj )) = 0, i.e. φn (zj ) and ψn (zj ) are linearly independent, i.e. φn (zj ) = bj ψn (zj )

(C.68)

for some complex constant bj . Then, from the asymptotics (C.12)–(C.13) φn (zj ) ∼ (zj )n

n → −∞ −n

φn (zj ) = bj ψn (zj ) ∼ bj (zj )

n → +∞

and since |zj | > 1 this means that zj is a proper eigenvalue. Vice-versa, proper eigenvalues outside the unit circle need to be zeros of a(z). Analogously one can show that z¯` with |¯ z` | < 1 is a proper eigenvalue if, and only if, a ¯(¯ z` ) = 0 and φ¯n (¯ z` ) = ¯b` ψ¯n (¯ zj ).

(C.69)

We will assume that neither a(z) nor a ¯(z) have zeros for |z| = 1. From the symmetries (C.59) and (C.63) it also follows a(z) = a ¯ (1/z) = a∗ (z ∗ ) a ¯(z) = a(1/z) = a ¯∗ (z ∗ )

(C.70) (C.71)

which means that if zj is an eigenvalue, so is zj∗ , i.e. the eigenvalues are real or come into complex conjugate pairs. Moreover, from (C.59) it follows that zj is an eigenvalue with |zj | > 1 if and only zj | < 1. Finally, the symmetry (C.66) yields that the eigenvalues if z¯j = z1j is an eigenvalue with |¯ also come in pairs ±zj (±¯ zj ). In conclusion, if the potential αn is real it has the following sets of oJ n . eigenvalues ±zj , ±zj∗ , ± z1j , ± z1∗ j

j=1

Let us write the scattering problem (C.3) in the form

wn+1 + wn−1 = −h2 k 2 pn wn where  −1 2 2 2 pn = α−1 = 1 + h |q | (1 + h2 |qn+1 | )−1 n n
˜ = z + z −1 = −h2 k 2 λ 146

(C.72) (C.73)

Then for real potentials we have the following relations 2

wn∗ (wn+1 + wn−1 ) = −h2 k 2 pn |wn |
∗ ∗ wn wn+1 + wn−1 = −h2 k ∗2 pn |wn |2

or, substracting one form another


2 ∗ ∗ wn + wn∗ wn−1 − wn+1 wn = −h2 k 2 − k ∗2 pn |wn | . wn∗ wn+1 − wn−1 Summing over all n at an eigenvalue kj yields kj2 − kj∗2

+∞
X

2

pn |wn (kj )| = 0

n=−∞

and, since 0 < pn < 1 for all n ∈ Z, if wn (zj ) ∈ L2 kj2 = kj∗2

(C.74)

˜ j = zj + z i.e. either kj is real or it is purely imaginary. This implies that λ is real and then eij ther zj is real or it is such that |zj | = 1. Since we assumed the potential to be such that a(z) 6= 0 for  J |z| = 1, we conclude that the discrete eigenvalues comes into quartets ±zj , ±zj−1 : zj ∈ (1, +∞) j=1 .
−1

C.2 C.2.1

Inverse Problem Recovery of the Jost functions

Suppose that a(z) has J simple zeros {zj : |zj | > 1}Jj=1 . Then the function µn (z) given by (C.52) is meromorphic outside the unite circle with a finite number of simple poles at the points z1 , . . . , zJ . Taking into account the symmetry (C.58), we can write eq. (C.50) as
¯n (z) + z −2n ρ(z)N ¯n z −1 µn (z) = N (C.75) which yields


¯n z −1 Cj Res (µn ; zj ) = lim (z − zj ) µn (z) = zj−2n N j z→zj

where

Cj = lim (z − zj ) ρ(z) = z→zj

b(zj ) a0 (zj )

(C.76)

(C.77)

are referred to as norming constants. We showed that the eigenvalues indeed come into pairs ±zj ; as far as the corresponding norming constants are concerned, we have the relation C−j =

b(−zj ) = −Cj a0 (−zj )

(C.78)

where we have taken into account that both a(z) and b(z) are even functions of the spectral parameter z (cf. (C.66)). Note that from (C.28) and (C.67) it follows that at large z µn (z) = ∆n + O(z −2 ) 147

(C.79)

where 

∆n = 

n−1 Y

j=−∞



 α−1 j

1−

+∞ X

(αk − 1)

k=−∞

k Y

`=−∞

α−1 `

!−1

.

(C.80)

Eq. (C.75) defines a Riemann-Hilbert boundary problem on the unit circle. However, from (C.79) it follows that the boundary conditions on µn depend on the potential, which is unknown in the inverse problem. Therefore, we introduce the following modified functions µn (z) ∆n ¯n (z) N ¯n0 (z) = N ∆n µ0n (z) =

(C.81) (C.82)

such that µ0n (z) = 1 + O(z −2 )
¯ 0 (z) = ∆−1 + O z 2 N n n

|z| → ∞ z → 0.

(C.83) (C.84)

¯n0 being properly defined, a small norm conditions on the poNote that, in order for µ0n and N tential αn is required to ensure that ∆n 6= 0 for all n ∈ Z and that it is well defined (i.e. P Qk −1 1 − +∞ k=−∞ (αk − 1) `=−∞ α` 6= 0). Eq. (C.75) then becomes
¯ 0 (z) + z −2n ρ(z)N ¯ 0 z −1 µ0n (z) = N (C.85) n n

and consequently


¯n0 z −1 Cj . Res (µ0n ; zj ) = zj−2n N j

Let us recall that we introduced the following projection operators I 1 f (w) P¯ (f )(z) = lim dw ζ→z 2πi |w|=1 w − ζ

(C.86)

(C.87)

|ζ|<1

which is defined for |z| ≤ 1 for any function f (z) continuous on |z| = 1 (we will refer to P¯ as to the inside projector), and I f (w) 1 dw (C.88) P (f )(z) = lim ζ→z 2πi |w|=1 w − ζ |ζ|>1

defined for |z| ≥ 1 for any function f (z) continuous on |z| = 1 (outside projector). Applying P¯ to both sides of eq. (C.85) and taking into account (C.83) and (C.86) yields
I J ¯n0 w−1 X zj−2n 0 −1
w−2n ρ(w)N 1 0 ¯ ¯ Nn (z) = 1 + Cj − N z lim dw z − zj n j 2πi ζ→z |w|=1 w−ζ j=1 |ζ|<1

or, equivalently, using (C.60) ¯ 0 (z) = 1 + N n

I J X ¯ 0 (w) zj−2n 0 −1
1 w2n−1 ρ¯(w)N n ¯ Nn zj Cj − lim dw. ζ→z |w|=1 z − z 2πi 1 − ζw j j=1 |ζ|<1

148

(C.89)

Note that the sum in the right-hand side is performed over all the discrete eigenvalues zj outside the unit circle (i.e. with |zj | > 1) and gives the contribution of the discrete spectrum, while the integral term corresponds to the
contribution of the continuous spectrum. Eq. (C.89) depends on ¯n0 z −1 , thefore, in order to close the system, we evaluate (C.89) at z −1 the additional constants N j ` for all ` = 1, . . . , J ¯n0 (z −1 ) = 1 + N `

J X j=1

zj−2n


¯n0 z −1 Cj − 1 N j −1 2πi z` − zj

I

|w|=1

¯n0 (w) w2n−1 ρ¯(w)N dw −1 1 − z` w

(C.90)

so that (C.89)–(C.90) constitute a linear algebraic-integral system on |z| = 1 which allows, in principle, given the scattering data {zj , Cj }Jj=1 ∪ {ρ(w) : |w| = 1} (i.e. eigenvalues and associated ¯ 0 (z). Then, norming constants and reflection coefficient) to recover the modified Jost functions N n taking into account the asymptotics (C.84), from (C.89) it follows ∆−1 n = 1−

J X


¯n0 z −1 Cj − 1 zj−2n−1 N j 2πi j=1

I

|w|=1

¯n0 (w) dw w2n−1 ρ¯(w)N

(C.91)

¯n (z) from N ¯n0 (z) (cf. (C.82)). which allows us to obtain N Note that for reflectionless potentials, i.e. when ρ(w) = 0 for all w such that |w| = 1, the system (C.89)-(C.90) reduces to a linear algebraic system. Eqs. (C.89)-(C.90) can be written so that one takes explicitly into account the symmetries. J  Indeed, we showed that the eigenvalues are real and come in quartets ±zj , ±zj−1 j=1 and for the correponding norming constants the relation (5.20) holds. Then the system (C.89)–(C.90) with the symmetries explicitly taken into account becomes ¯ 0 (z) = 1 + N n

J X

zj−2n

j=1

1 − lim 2πi ζ→z

|ζ|<1

J X
¯n0 z −1 = 1 + zj−2n N ` j=1

I




1 ¯0 1 ¯ 0 −1
−1 − N z N −zj Cj z − zj n j z + zj n

|w|=1



w2n−1 ρ¯(w)N¯n0 (w) dw 1 − ζw



1 1 −1 −1 0 0 ¯ ¯ Cj Nn zj − −1 Nn −zj z`−1 − zj z` + zj

¯n0 (w) w2n−1 ρ¯(w)N dw 1 − z`−1 w |w|=1   J X


1 1 −2n ¯n0 −z −1 = 1 + ¯n0 z −1 − ¯n0 −z −1 Cj z N N N j j j ` −z`−1 − zj −z`−1 + zj j=1 I ¯n0 (w) 1 w2n−1 ρ¯(w)N − dw −1 2πi |w|=1 1 + z` w
¯ 0 ±z −1 for ` = 1, . . . , J) where J is the and this system of 2J equations in 2J unknowns (N n ` number of eigenvalues in the interval (1, +∞) admits the symmetry 1 − 2πi

I

¯n0 (z) = N ¯n0 (−z) N

149

so that it reduces to ¯ 0 (z) = 1 + 2 N n

I J X ¯ 0 (w) zj−2n+1 0 −1
1 w2n−1 ρ¯(w)N n ¯ z N C − lim dw j j 2 − z2 n ζ→z |w|=1 z 2πi 1 − ζw j j=1

(C.92)

|ζ|<1

J X
zj−2n+1 0 −1
1 ¯n0 z −1 = 1 + 2 ¯n z Cj − N N j ` −2 2 2πi z − zj j=1 `

I

¯n0 (w) w2n−1 ρ¯(w)N dw 1 − z`−1 w

(C.93)

I

¯n0 (w) dw. w2n−1 ρ¯(w)N

(C.94)

|w|=1

and eq. (C.91) becomes ∆−1 n =1−2

C.2.2

J X


¯n0 z −1 Cj − 1 zj−2n−1 N j 2πi j=1

|w|=1

Recovery of the potential

¯n (z) is analytic inside the unit circle, it admits a convergent Taylor series Since the eigenfunction N about z = 0. Hence, let us write ¯n (z) = 1 + z 2 N ¯ (2) + . . . N n

(C.95)

¯n is an even function of the spectral parameter z (cf. where we have taken into account that N (C.26)). Substituting this expansion into the scattering equation (C.16) and equating the powers of z we find (2)

¯ (2) − N ¯ (αn − 1) = N n n−1

O(z) :

(C.96)

which allows us to reconstruct the potential in terms of the coefficients of the Taylor seris expansion ¯n (z), namely of the eigenfunction N ¯ (2) − N ¯ (2) . αn = 1 + N n n−1

(C.97)

Reflectionless potentials In the reflectionless case, (C.92)–(C.94) reduce ¯n0 (z) = 1 + 2 N

J X zj−2n+1 0 −1
¯ z N Cj j 2 − z2 n z j j=1

J X
zj−2n+1 0 −1
¯n0 z −1 = 1 + 2 ¯ z Cj N N j ` −2 2 n z − z j j=1 `

∆−1 n = 1−2

J X j=1


¯ 0 z −1 Cj zj−2n−1 N n j

and (C.82) gives ¯n (z) ∼ ∆n + ∆n N ¯n0(2) z 2 + . . . N

150

j = 1, . . . , J

(C.98)

(C.99)

(C.100)

which implies J X


¯n z −1 Cj . zj−2n−3 N j

(C.101)



 ¯n−1 z −1 − z −2 N ¯n z −1 zj−2n−1 N j j j

(C.102)

¯ (2) = −2 N n

j=1

Therefore, taking into account (C.97) αn = 1 + 2

J X j=1

For J = 1 the eqs. (C.98)–(C.99) give
¯ 0 z −1 = N n 1 ∆−1 n =

1 z12

1+2

1 + 2C1 1 + 2C1

therefore from (C.82) we get
¯n z −1 = N 1


−1 z1−2

C1 z1−2n+1
−1 −2n−3 z12 − z1−2 z1
−1 z1−2n+1 z12 − z1−2 −

1 1+

2C1 z12

− z1−2

and then from (C.102) it follows αn = 1 + h 1 + 2C1


−1

z1−2n−3


2C1 1 − z1−2 z1−n−1
−1 −2n−2 i h
−1 −2n−3 i . z12 − z1−2 1 + 2C1 z12 − z1−2 z1 z1

(C.103)

In order to get a real and nonsingular potential, the norming constant C1 has to be real and of the same sign as z1 , i.e. C1 > 0 if z1 ∈ (1, +∞). Let us write z1 in the form z1 = eω

ω > 0.

Then the potential becomes

or


2C1 e−(2n+1)ω − e−(2n+3)ω   αn = 1 +  C1 C1 −(2n+1)ω −(2n+3)ω 1 + sinh 1 + sinh 2ω e 2ω e αn = 1 +

where

cosh



n+

e

x0

1 2

=




sinh ω sinh 2ω  
 ω + x0 cosh n + 23 ω + x0



sinh 2ω C1

151

1/2

.

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