Inverse scattering transform for the integrable discrete nonlinear Schrödinger equation with nonvanishing boundary conditions

Ablowitz, Mark J.; Biondini, Gino; Prinari, B.

doi:10.1088/0266-5611/23/4/021

Cited by 99 publications

(156 citation statements)

References 22 publications

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“…(A.5) and Appendix A.] Hence, under the same regularity hypotheses as before, Φ (1) n , Φ (2) n and Φ (3) n are each fundamental solutions of the Lax pair (A.5). We can therefore write the following relations among the modified eigenfunctions: µ…”

Section: The Ablowitz-ladik System On the Naturalsmentioning

confidence: 86%

“…It is then easy to define φ (1,2) n (z, t) as the solutions of (2.6) which vanish as n → ∓∞, respectively:…”

Section: Ivp For Dls Via Spectral Analysis Of the Lax Pairmentioning

confidence: 99%

“…Moreover, let A(z, T 0 ) and B(z, T 0 ) be the spectral coefficients obtained by replacing T with T 0 in (E.3), and letJ (1) n (z, t), . .…”

Section: Appendix a Notation And Frequently Used Formulaementioning

confidence: 99%

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Initial-boundary-value problems for discrete evolution equations: discrete linear Schrödinger and integrable discrete nonlinear Schrödinger equations

Biondini

Hwang

2008

Inverse Problems

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Abstract. We present a method to solve initial-boundary value problems for linear and integrable nonlinear differential-difference evolution equations. The method is the discrete version of the one developed by A. S. Fokas to solve initial-boundary value problems for linear and integrable nonlinear partial differential equations via an extension of the inverse scattering transform. The method takes advantage of the Lax pair formulation for both linear and nonlinear equations, and is based on the simultaneous spectral analysis of both parts of the Lax pair. A key role is also played by the global algebraic relation that couples all known and unknown boundary values. Even though additional technical complications arise in discrete problems compared to continuum ones, we show that a similar approach can also solve initial-boundary value problems for linear and integrable nonlinear differential-difference equations. We demonstrate the method by solving initial-boundary value problems for the discrete analogue of both the linear and the nonlinear Schrödinger equations, comparing the solution to those of the corresponding continuum problems. In the linear case we also explicitly discuss Robin-type boundary conditions not solvable by Fourier series. In the nonlinear case we also identify the linearizable boundary conditions, we discuss the elimination of the unknown boundary datum, we obtain explicitly the linear and continuum limit of the solution, and we write down the soliton solutions.28 December 2017

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Section: The Ablowitz-ladik System On the Naturalsmentioning

confidence: 86%

“…It is then easy to define φ (1,2) n (z, t) as the solutions of (2.6) which vanish as n → ∓∞, respectively:…”

Section: Ivp For Dls Via Spectral Analysis Of the Lax Pairmentioning

confidence: 99%

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Initial-boundary-value problems for discrete evolution equations: discrete linear Schrödinger and integrable discrete nonlinear Schrödinger equations

Biondini

Hwang

2008

Inverse Problems

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“…Here we would like to stress that sometimes the integration even of simplest nonlinear systems associated with the second order scattering problems turns out to be very difficult task due to the complications inflicted by the nonvanishing boundary conditions for the field variables. As an example it is sufficient to mention the situation concerning the famous Ablowitz-Ladik system [9][10][11] when the solutions with nonvanishing boundary conditions [19,20] have been found well after the solutions with the vanishing ones [10,11]. Whether the inverse scattering theory developed by Beals and Coifman for the continuous space variable [21] or by Bhate for the discrete space variable [22] could provide one with a procedure suitable to integrate our semidiscrete systems in more simple and straightforward manner as compared with the Caudrey scheme [6][7][8] will be checked by time.…”

Section: Discussionmentioning

confidence: 99%

Semidiscrete Integrable Nonlinear Systems Generated by the New Fourth Order Spectral Operator: Systems of Obverse Type

Vakhnenko¹

2021

JNMP

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In the framework of zero-curvature representation we have proposed three distinct versions of semidiscrete integrable nonlinear systems arising due to a proper multifield augment of integrable nonlinear Schrödinger system with the background-controlled intersite resonant couplings. The specification either of these three systems is essentially based upon the lowest local conservation laws early found by means of modified recurrence procedure and consists in a proper fixation of sampling functions within the general evolution operator of obverse type. The number of actual field variables in each of obtained systems is shown to be considerably reduced due to the two natural constraints independent of sampling fixation and two additional constraints dictated by the chosen sampling.

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“…In [6,24] the matrix equations (1.1a), (1.1b) were studied in detail using the IST. In [1,25] the defocussing N = M = 1 problem was studied for potentials not vanishing as n → ±∞.…”

Section: Introductionmentioning

confidence: 99%

An Alternative Approach to Integrable Discrete Nonlinear Schrödinger Equations

Demontis

Mee

2013

Acta Appl Math

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In this article we develop the direct and inverse scattering theory of a discrete matrix Zakharov-Shabat system with solutions U n and W n . Contrary to the discretization scheme enacted by Ablowitz and Ladik, a central difference scheme is applied to the positional derivative term in the matrix Zakharov-Shabat system to arrive at a different discrete linear system. The major effect of the new discretization is that we no longer need the following two conditions in theories based on the Ablowitz-Ladik discretization: (a) invertibility of I N − U n W n and I M − W n U n , and (b) I N − U n W n and I M − W n U n being nonzero multiples of the respective identity matrices I N and I M .

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Inverse scattering transform for the integrable discrete nonlinear Schrödinger equation with nonvanishing boundary conditions

Cited by 99 publications

References 22 publications

Initial-boundary-value problems for discrete evolution equations: discrete linear Schrödinger and integrable discrete nonlinear Schrödinger equations

Initial-boundary-value problems for discrete evolution equations: discrete linear Schrödinger and integrable discrete nonlinear Schrödinger equations

Semidiscrete Integrable Nonlinear Systems Generated by the New Fourth Order Spectral Operator: Systems of Obverse Type

An Alternative Approach to Integrable Discrete Nonlinear Schrödinger Equations

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