# Classification of a Subclass of Two-Dimensional Lattices via Characteristic Lie Rings

@article{Habibullin2017ClassificationOA,
title={Classification of a Subclass of Two-Dimensional Lattices via Characteristic Lie Rings},
author={Ismagil Talgatovich Habibullin and M. N. Poptsova},
journal={Symmetry Integrability and Geometry-methods and Applications},
year={2017},
volume={13},
pages={073}
}
• Published 29 March 2017
• Mathematics
• Symmetry Integrability and Geometry-methods and Applications
The main goal of the article is testing a new classification algorithm. To this end we apply it to a relevant problem of describing the integrable cases of a subclass of two-dimensional lattices. By imposing the cut-off conditions $u_{-1}=c_0$ and $u_{N+1}=c_1$ we reduce the lattice $u_{n,xy}=\alpha(u_{n+1},u_n,u_{n-1})u_{n,x}u_{n,y}$ to a finite system of hyperbolic type PDE. Assuming that for each natural $N$ the obtained system is integrable in the sense of Darboux we look for $\alpha$. To…

### Algebraic properties of quasilinear two-dimensional lattices connected with integrability

• Mathematics
• 2018
In the article a classification method for nonlinear integrable equations with three independent variables is discussed based on the notion of the integrable reductions. We call the equation

### Integrability conditions for two-dimensional lattices

• Mathematics
• 2020
In the article some algebraic properties of nonlinear two-dimensional lattices of the form $u_{n,xy} = f(u_{n+1}, u_n, u_{n-1})$ are studied. The problem of exhaustive description of the integrable

### Classification of a subclass of quasilinear two-dimensional lattices by means of characteristic algebras

We consider a classification problem of integrable cases of the Toda type twodimensional lattices un,xy = f(un+1, un, un−1, un,x, un,y). The function f = f(x1, x2, · · ·x5) is assumed to be analytic

### A classification algorithm for integrable two-dimensional lattices via Lie—Rinehart algebras

• Mathematics
• 2020
We study the problem of the integrable classification of nonlinear lattices depending on one discrete and two continuous variables. By integrability, we mean the presence of reductions of a chain to

### On a class of 2D integrable lattice equations

• Mathematics
• 2020
We develop a new approach to the classification of integrable equations of the form $$u_{xy}=f(u, u_x, u_y, \triangle_z u \triangle_{\bar z}u, \triangle_{z\bar z}u),$$ where $\triangle_{ z}$ and

### Lax pair for one novel two-dimensional lattice

In our recent papers [1, 2] the algorithm for classification of integrable equations with three independent variables was proposed. This method is based on the requirement of the existence of an

### Characteristic Lie algebras of integrable differential-difference equations in 3D

• Mathematics
Journal of Physics A: Mathematical and Theoretical
• 2021
The purpose of this article is to discuss an algebraic method for studying integrable differential-difference lattices with two discrete and one continuous independent variables. The main idea is

### Lax Pair for a Novel Two-Dimensional Lattice

• M. N. Kuznetsova
• Mathematics
Symmetry, Integrability and Geometry: Methods and Applications
• 2021
In paper by I.T. Habibullin and our joint paper the algorithm for classification of integrable equations with three independent variables was proposed. This method is based on the requirement of the

### Integrability conditions for two-dimensional Toda-like equations

• Mathematics
Journal of Physics A: Mathematical and Theoretical
• 2020
In the article some algebraic properties of nonlinear two-dimensional lattices of the form un,xy = f(un+1, un, un−1) are studied. The problem of exhaustive description of the integrable cases of this

### Dedicated to the blessed memory of our teachers and senior colleagues ,

• Mathematics
• 2021
In the present paper we study characteristic algebras for exponential systems corresponding to degenerate Cartan matrices. These systems generalize hyperbolic sineGordon and Tzitzeica equations

## References

SHOWING 1-10 OF 32 REFERENCES

### Darboux integrability of trapezoidal $H^{4}$ and $H^{6}$ families of lattice equations II: General Solutions

• Mathematics
• 2017
In this paper we construct the general solutions of two families of partial difference equations defined on the quad graph, namely the trapezoidal $H^4$ equations and the $H^6$ equations. These

### On a Class of Three-Dimensional Integrable Lagrangians

• Mathematics, Physics
• 2004
AbstractWe characterize non-degenerate Lagrangians of the form such that the corresponding Euler-Lagrange equations are integrable by the method of hydrodynamic reductions. The integrability

### Characteristic Lie rings, finitely-generated modules and integrability conditions for (2 + 1)-dimensional lattices

Characteristic Lie rings for Toda and Volterra type (2 + 1)-dimensional lattices are defined. Some properties of these rings are studied. Infinite sequence of special kind subrings are introduced. It

### Grassmannians Gr(N − 1, N + 1), closed differential N − 1-forms and N-dimensional integrable systems

• Mathematics
• 2013
Integrable flows on the Grassmannians Gr(N − 1, N + 1) are defined by the requirement of closedness of the differential N − 1-forms ΩN − 1 of rank N − 1 naturally associated with Gr(N − 1, N + 1).

### On the Integrability of (2+1)-Dimensional Quasilinear Systems

• Mathematics
• 2004
A (2+1)-dimensional quasilinear system is said to be ‘integrable’ if it can be decoupled in infinitely many ways into a pair of compatible n-component one-dimensional systems in Riemann invariants.

### Integrable (2+1)-dimensional systems of hydrodynamic type

• Mathematics
• 2010
We describe the results that have so far been obtained in the classification problem for integrable (2+1)-dimensional systems of hydrodynamic type. The Gibbons-Tsarev (GT) systems are most

### The Symmetry Approach to Classification of Integrable Equations

• Mathematics
• 1991
In this volume each of the contributors proposes his own test to recognize integrable PDEs. We believe that, independently from the basic definition of integrability, the test must satisfy some

### Towards classification of -dimensional integrable equations. Integrability conditions I

• Mathematics
• 1998
In this paper we attempt to extend the symmetry approach (well developed in the case of (1 + 1)-dimensional equations) to the (2 + 1)-dimensional case. Presence of nonlocal terms in symmetries and

### On a class of Darboux-integrable semidiscrete equations

• Mathematics
• 2017
We consider a classification problem for Darboux-integrable hyperbolic semidiscrete equations. In particular, we obtain a complete description for a special class of equations admitting

### Symmetry Approach to the Integrability Problem

• Mathematics
• 2000
We review the results of the twenty-year development of the symmetry approach to classifying integrable models in mathematical physics. The generalized Toda chains and the related equations of the