On the classification of nonlinear integrable three-dimensional chains by means of characteristic Lie algebras

2023/06/26 by I. T. Habibullin, Habibullin, I T, A R Khakimova +1
Physics and Astronomy · #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2306.14585

openalex publication_date 2023/06/26 · openalex created_date 2023/06/28 · openalex updated_date 2026/07/28

Abstract

The article continues the work on the description of integrable nonlinear chains with three independent variables of the following form u^j_n+1,x=u^j_n,x+f(u^j+1_n, u^j_n,u^j_n+1 ,u^j-1_n+1) by the presence of a hierarchy of reductions integrable in the sense of Darboux, started in (1). The classification algorithm is based on the well-known fact that the characteristic algebras of Darboux integrable systems have a finite dimension. In this paper, we used the characteristic algebra in the direction x, whose structure for a given class of models is determined by some polynomial P(λ), whose degree does not exceed three for known examples. The article assumes that P(λ)=λ², in this case the classification problem is reduced to finding eight unknown functions of one variable. In the paper, a rather narrow class of candidates for integrability is obtained, among which there is a new example of an integrable chain.

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