On the $3D$ consistency of a Grassmann extended lattice Boussinesq system

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

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 integra…

关于 $3 \times 3\times 3$ 立方体周围的一致性及 Q3 类型的格点 Boussinesq 方程类比

On consistency around a $3 \times 3\times 3$ cube and Q3 analogue of the lattice Boussinesq equation

本文提出了一种扩展三维一致性的格点 BSQ 方程，并构造了 Q3 类型的格点 BSQ-Q3 系统，作为 BSQ 类比的 Q3($δ$) 方程的推广。

Lattice Schwarzian Boussinesq equation and two-component systems

Various new two-component systems related to the lattice Schwarzian Boussinesq equation are constructed in a systematic way from conservation laws. Their multidimensional consistency is demonstrated, Lax pairs, symmetries and conservation laws are derived and an auto-Backlund transformation is constructed. Finally, Yang-Baxter maps from these systems are constructed.

Soliton Taxonomy for a Modification of the Lattice Boussinesq Equation

Integrable multi-component lattice equations of the Boussinesq family have been known for some time. Recently some new equations of this type were found using the Consistency-Around-the-Cube approach. Here we investigate one of these models, B-2, and in particular the consequences of a nonzero deformation parameter $b_0>0$, which allows special kinds of solitons in the parameter range $-b_0/3<k<b_0$.

Multisoliton solutions to the lattice Boussinesq equation

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 $[(a-ωk)/(a-k)]^n$ and $[(a-ω^2 k)/(a-k)]^n$, where $ω$ is a cubic root of unity $\neq 1$.

Non-commutative rational Yang-Baxter maps

Starting from multidimensional consistency of non-commutative lattice modified Gel'fand-Dikii systems we present the corresponding solutions of the functional (set-theoretic) Yang-Baxter equation, which are non-commutative versions of the maps arising from geometric crystals. Our approach works under additional condition of centrality of certain products of non-commuting variables. Then we apply such a restriction on the level of the Gel'fand-Dikii systems what allows to obtain non-autonomous (but with central non-autonomous factors) versions of the equations. In particular we recover known no…