Boussinesq-like multi-component lattice equations and multi-dimensional consistency

TL;DR

This paper classifies and analyzes multi-component lattice equations inspired by the Boussinesq equation, demonstrating their multidimensional consistency and deriving new integrable equations, some of which generalize the lattice modified Boussinesq equation.

Contribution

The paper introduces a classification of multi-component lattice equations based on their multidimensional consistency and derives new integrable equations generalizing the lmBSQ.

Findings

Classification into three canonical forms.

Derivation of consistency conditions and their solutions.

Identification of new integrable multi-component lattice equations.

Abstract

We consider quasilinear, multi-variable, constant coefficient, lattice equations defined on the edges of the elementary square of the lattice, modeled after the lattice modified Boussinesq (lmBSQ) equation, e.g., $\tilde y z=\tilde x-x$. These equations are classified into three canonical forms and the consequences of their multidimensional consistency (Consistency-Around-the-Cube, CAC) are derived. One of the consequences is a restriction on form of the equation for the $z$ variable, which in turn implies further consistency conditions, that are solved. As result we obtain a number of integrable multi-component lattice equations, some generalizing lmBSQ.