Hirota's method and the search for integrable partial difference equations. 1. Equations on a 3x3 stencil

TL;DR

This paper explores the use of Hirota's bilinear method to identify integrable partial difference equations on a 3x3 grid, linking them to higher-dimensional master equations and emphasizing the role of three-soliton solutions in determining integrability.

Contribution

It applies Hirota's method to 2D partial difference equations on a 3x3 stencil, establishing criteria for integrability based on three-soliton solutions and connecting these equations to Hirota-Miwa master equations.

Findings

Three-soliton solutions characterize integrability.

Certain equations are derived as limits of Hirota-Miwa equations.

Singular limits are sometimes necessary to relate equations.

Abstract

Hirota's bilinear method ("direct method") has been very effective in constructing soliton solutions to many integrable equations. The construction of one- and two-soliton solutions is possible even for non-integrable bilinear equations, but the existence of a generic three-soliton solution imposes severe constraints and is in fact equivalent to integrability. This property has been used before in searching for integrable partial differential equations, and in this paper we apply it to two dimensional partial difference equations defined on a 3x3 stencil. We also discuss how the obtained equations are related to projections and limits of the three-dimensional master equations of Hirota and Miwa, and find that sometimes a singular limit is needed.