Soliton Solutions for ABS Lattice Equations II: Casoratians and Bilinearization

TL;DR

This paper develops a new method for deriving soliton solutions to ABS lattice equations using Hirota's direct approach, resulting in Casoratians and bilinear equations, expanding on previous work that used Cauchy matrices.

Contribution

It introduces a Hirota-based method for constructing soliton solutions to ABS lattice equations, providing explicit Casoratian and bilinear forms for H-series and Q1 equations.

Findings

Derived N-soliton solutions using Hirota's method

Established Casoratian and bilinear difference equations

Extended results to H-series and Q1 equations

Abstract

In Part I [arXiv:0902.4873 [nlin.SI]] soliton solutions to the ABS list of multi-dimensionally consistent difference equations (except Q4) were derived using connection between the Q3 equation and the NQC equations, and then by reductions. In that work central role was played by a Cauchy matrix. In this work we use a different approach, we derive the $N$-soliton solutions following Hirota's direct and constructive method. This leads to Casoratians and bilinear difference equations. We give here details for the H-series of equations and for Q1; the results for Q3 have been given earlier.