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

TL;DR

This paper extends Fokas' unified transform method to discrete evolution equations, enabling the solution of initial-boundary value problems for discrete linear and nonlinear Schrödinger equations using spectral analysis and Lax pairs.

Contribution

It develops a discrete analogue of Fokas' method for solving initial-boundary value problems, including handling boundary conditions and soliton solutions for integrable discrete equations.

Findings

Successfully solves initial-boundary value problems for discrete Schrödinger equations.

Identifies linearizable boundary conditions and explicit solutions.

Connects discrete solutions to continuum limits and soliton structures.

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 Schrodinger 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.