# Discrete Crum's Theorems and Integrable Lattice Equations

## TL;DR

This paper develops discrete Darboux transformations and Crum's theorems for second order difference equations related to integrable systems, leading to explicit multi-soliton solutions for lattice KdV, mKdV, and sine-Gordon equations.

## Contribution

It introduces discrete versions of Crum's theorems for difference equations, enabling construction of integrable lattice equations and their multi-soliton solutions.

## Key findings

- Discrete Crum's theorems for second order difference equations
- Explicit multi-soliton solutions for lattice integrable equations
- Unified construction of continuous, semi-discrete, and lattice integrable systems

## Abstract

In this paper, we develop discrete versions of Darboux transformations and Crum's theorems for two second order difference equations. The difference equations are discretised versions (using Darboux transformations) of the spectral problems of the KdV quation, and of the modified KdV equation or sine-Gordon equation. Considering the discrete dynamics created by Darboux transformations for the difference equations, one obtains the lattice potential KdV equation, the lattice potential modified KdV equation and the lattice Schwarzian KdV equation, that are prototypes of integrable lattice equations. It turns out that, along the discretisation processes using Darboux transformations, two families of integrable systems (the KdV family, and the modified KdV or sine-Gordon family), including their continuous, semi-discrete and lattice versions, are explicitly constructed. As direct applications of the discrete Crum's theorems, multi-soliton solutions of the lattice equations are obtained.