Binary Darboux transformations for discrete modified Boussinesq equation

TL;DR

This paper derives the two-component discrete modified Boussinesq equation and its Lax pair using a 3-periodic reduction, then constructs Darboux transformations to generate explicit solutions including N-solitons.

Contribution

It introduces a novel 3-periodic reduction method for the Hirota-Miwa equation and develops Darboux transformations for the discrete modified Boussinesq equation.

Findings

Derived the two-component discrete modified Boussinesq equation.

Constructed Darboux and binary Darboux transformations for the equation.

Obtained explicit N-soliton solutions using Casorati- and Gramm-type determinants.

Abstract

We obtain the well-known discrete modified Boussinesq equation in two-component form as well as its Lax pair in $3\times3$ matrix form through a 3-periodic reduction technique on the Hirota-Miwa equation and its Lax pair. We describe how Darboux transformations and binary Darboux transformations can be constructed for this two-component discrete integrable equation. These transformations are then used to obtain classes of explicit solutions in the form of Casorati- and Gramm-type determinants. $N$-soliton solutions of the discrete modified Boussinesq equation are discussed as well when taking the vacuum potentials as constants.