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

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.

24 pages

References in corpus (3)

• Classification of integrable equations on quad-graphs. The consistency approach
• Lax pair for the Adler (lattice Krichever-Novikov) System
• The discrete potential Boussinesq equation and its multisoliton solutions

Cited by in corpus (23)

• Symbolic Computation of Lax Pairs of Partial Difference Equations Using Consistency Around the Cube
• Soliton Taxonomy for a Modification of the Lattice Boussinesq Equation
• An integrable multicomponent quad equation and its Lagrangian formulation
• Non-commutative lattice modified Gel'fand-Dikii systems
• Multi-Component Extension of CAC Systems
• Rational Solutions to the ABS List: Transformation Approach
• Symmetries and conservation laws of lattice Boussinesq equations
• Non-Abelian hierarchies of compatible maps, associated integrable difference systems and Yang-Baxter maps
• A higher-rank version of the Q3 equation
• Direct linearisation approach to discrete integrable systems associated with ${\mathbb{Z}}_{\mathcal{N}}$ graded Lax pairs
• Asymptotic diagonalization of the Discrete Lax pair around singularities and conservation laws for dynamical systems
• Conservation laws of some lattice equations
• Kadomtsev-Petviashvili system and reduction: generalized Cauchy matrix approach
• Discrete Lax pairs and hierarchies of integrable difference systems
• Degree growth of lattice equations defined on a 3x3 stencil
• Lax matrices for lattice equations which satisfy consistency-around-a-face-centered-cube
• On the Lagrangian multiform structure of the extended lattice Boussinesq system
• Direct Linearization of extended lattice BSQ systems
• Spectrum transformation and conservation laws of the lattice potential KdV equation
• Binary Darboux transformations for discrete modified Boussinesq equation
• Elliptic solutions of Boussinesq type lattice equations and the elliptic Nth root of unity
• Classification of integrable boundary equations for integrable quad-graph systems
• Search for integrable two-component versions of the lattice equations in the ABS-list