Integrable Discretization of Soliton Equations via Bilinear Method and B\"acklund Transformation

TL;DR

This paper develops a systematic method using Hirota's bilinear approach and B"acklund transformations to derive integrable discrete versions of several well-known PDEs, ensuring their continuum limits recover the original equations.

Contribution

It introduces a unified procedure for discretizing integrable PDEs while preserving their integrability properties, including B"acklund transformations and Lax pairs.

Findings

Derived semi-discrete analogues of several extended integrable equations.

Established continuum limits that recover original PDEs.

Provided explicit B"acklund transformations and Lax pairs for new systems.

Abstract

In this paper, we present a systematic procedure to derive discrete analogues of integrable PDEs via Hirota's bilinear method. This approach is mainly based on the compatibility between an integrable system and its B\"acklund transformation. We apply this procedure to several equations, including the extended Korteweg-de-Vries (KdV) equation, the extended Kadomtsev-Petviashvili (KP) equation, the extended Boussinesq equation, the extended Sawada-Kotera (SK) equation and the extended Ito equation, and obtain their associated semi-discrete analogues. In the continuum limit, these differential-difference systems converge to their corresponding smooth equations. For these new integrable systems, their B\"acklund transformations and Lax pairs are derived