Multisoliton solutions to the lattice Boussinesq equation

TL;DR

This paper derives multisoliton solutions for the lattice Boussinesq equation using Hirota bilinear form and Casoratians, revealing a novel discretization involving cubic roots of unity.

Contribution

It introduces a new method for constructing multisoliton solutions of the lattice Boussinesq equation with a unique discretization approach involving cubic roots of unity.

Findings

Multisoliton solutions expressed via Casoratians.

A novel discretization involving two different exponential terms.

Extension of Hirota bilinear method to the lattice Boussinesq equation.

Abstract

The lattice Boussinesq equation (BSQ) is a three-component difference-difference equation defined on an elementary square of the 2D lattice, having 3D consistency. We write the equations in the Hirota bilinear form and construct their multisoliton solutions in terms of Casoratians, following the methodology in our previous papers. In the construction it turns out that instead of the usual discretization of the exponential as $[(a+k)/(a-k)]^n$ we need two different terms $[(a-\omega k)/(a-k)]^n$ and $[(a-\omega^2 k)/(a-k)]^n$, where $\omega$ is a cubic root of unity $\neq 1$.