# Multidimensional inverse scattering of integrable lattice equations

@article{Butler2012MultidimensionalIS, title={Multidimensional inverse scattering of integrable lattice equations}, author={Samuel Butler}, journal={Nonlinearity}, year={2012}, volume={25}, pages={1613 - 1634} }

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…

## 10 Citations

### Spectrum transformation and conservation laws of lattice potential KdV equation

- Mathematics
- 2015

Many multi-dimensional consistent discrete systems have soliton solutions with nonzero backgrounds, which brings difficulty in the investigation of integrable characteristics. In this paper, we…

### Solutions to ABS Lattice Equations via Generalized Cauchy Matrix Approach

- Mathematics
- 2012

The usual Cauchy matrix approach starts from a known plain wave factor vector r and known dressed Cauchy matrix M . In this paper, we start from a determining matrix equation set with undetermined r…

### A Discrete Inverse Scattering Transform for Q3$_\delta$

- Mathematics
- 2012

We derive a fully discrete Inverse Scattering Transform as a method for solving the initial-value problem for the Q3$_\delta$ lattice (difference-difference) equation for real-valued solutions. The…

### Algebro-geometric solutions to the lattice potential modified Kadomtsev–Petviashvili equation

- MathematicsJournal of Physics A: Mathematical and Theoretical
- 2022

Algebro-geometric solutions of the lattice potential modified Kadomtsev–Petviashvili (lpmKP) equation are constructed. A Darboux transformation of the Kaup–Newell spectral problem is employed to…

### Darboux and binary Darboux transformations for discrete integrable systems I. Discrete potential KdV equation

- Mathematics
- 2013

The Hirota–Miwa equation can be written in ‘nonlinear’ form in two ways: the discrete KP equation and, by using a compatible continuous variable, the discrete potential KP equation. For both systems,…

### On one-soliton solutions of the Q2 equation in the ABS list

- PhysicsAdvances in Difference Equations
- 2019

In this paper, we derive seed and 1-soliton solutions of the Q2 equation in the Adler–Bobenko–Suris list. The seed solutions of Q2 are obtained using those of Q1(δ)$\mbox{Q}1(\delta)$ and an non-auto…

### The Sylvester equation and the elliptic Korteweg-de Vries system

- Computer Science
- 2015

The elliptic Korteweg-de Vries (KdV) system is a multi-component generalization of the lattice potential KdV equation, whose soliton solutions are associated with an elliptic Cauchy kernel (i.e., a…

### Rational solutions to Q3δ in the Adler-Bobenko-Suris list and degenerations

- Physics, MathematicsJournal of Nonlinear Mathematical Physics
- 2018

We derive rational solutions in Casoratian form for the Nijhoff-Quispel-Capel (NQC) equation by using the lattice potential Korteweg-de Vries (lpKdV) equation and two Miura transformations between…

### Rational solutions to the ABS list: Degenerating approach

- Mathematics
- 2017

In the paper we first construct rational solutions for the Nijhoff-Quispel-Capel (NQC) equation by means of bilinear method. These solutions can be transferred to those of Q3$_\delta$ equation in the…

### Inverse Scattering Transform Method for Lattice Equations

- Mathematics
- 2012

Submitted in fulfilment of the requirements for the degree of Doctor of Philosophy to the School of Mathematics and Statistics, Faculty of Science.

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