Bilinear Equations and B\"acklund Transformation for Generalized Ultradiscrete Soliton Solution

TL;DR

This paper introduces ultradiscrete soliton equations and B"acklund transformations for generalized solutions, expressing solutions via ultradiscrete permanents and exploring their relation to discrete cases.

Contribution

It presents a generalized ultradiscrete soliton solution, including equations like ultradiscrete KdV and Toda, and connects B"acklund transformations across discrete and ultradiscrete frameworks.

Findings

Ultradiscrete soliton equations are formulated with B"acklund transformations.

Solutions are expressed using ultradiscrete permanents.

The relation between discrete and ultradiscrete B"acklund transformations is analyzed.

Abstract

Ultradiscrete soliton equations and B\"acklund transformation for a generalized soliton solution are presented. The equations include the ultradiscrete KdV equation or the ultradiscrete Toda equation in a special case. We also express the solution by the ultradiscrete permanent, which is defined by ultradiscretizing the signature-free determinant, that is, the permanent. Moreover, we discuss a relation between B\"acklund transformations for discrete and ultradiscrete KdV equations.