Hirota difference equation: IST, Darboux transformation and solitons

TL;DR

This paper explores the Hirota difference equation by analyzing its direct and inverse problems, introducing Jost solutions and scattering data, and demonstrating Darboux transformations for soliton solution construction.

Contribution

It provides a detailed analysis of the Hirota difference equation, including the development of Darboux transformations and a recursive method for soliton solutions.

Findings

Darboux transformation induces discrete time evolution.

Recursion procedure for Jost solutions at arbitrary times.

Properties of soliton solutions are characterized.

Abstract

Direct and inverse problems for the Hirota difference equation are considered. Jost solutions and scattering data are introduced and their properties are presented. Darboux transformation in a special case is shown to give evolution with respect to discrete time and a recursion procedure for consequent construction of the Jost solution at arbitrary time, if the initial value is given. Some properties of the soliton solutions are discussed.