Multidimensional Inverse Scattering of Integrable Lattice Equations

TL;DR

This paper develops a multidimensional inverse scattering transform for a broad class of integrable lattice equations, enabling the analysis of solutions in higher-dimensional discrete systems.

Contribution

It introduces a novel multidimensional inverse scattering method for all ABS equations except Q4, expanding the analytical tools for integrable lattice equations.

Findings

Derived soliton solutions via Cauchy matrix approach

Established equivalence between soliton solutions and reflectionless potentials

Discussed inverse scattering solutions for lattice KdV and related equations

Abstract

We present a discrete inverse scattering transform for all ABS equations excluding Q4. The nonlinear partial difference equations presented in the ABS hierarchy represent a comprehensive class of scalar affine-linear lattice equations which possess the multidimensional consistency property. Due to this property it is natural to consider these equations living in an N-dimensional lattice, where the solutions depend on N distinct independent variables and associated parameters. The direct scattering procedure, which is one-dimensional, is carried out along a staircase within this multidimensional lattice. The solutions obtained are dependent on all N lattice variables and parameters. We further show that the soliton solutions derived from the Cauchy matrix approach are exactly the solutions obtained from reflectionless potentials, and we give a short discussion on inverse scattering solutions of some previously known lattice equations, such as the lattice KdV equation.