On the classification of nonlinear integrable three-dimensional chains by means of characteristic Lie algebras
arXiv (Cornell University)(2023)
摘要
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 $P(\lambda)$, whose degree does not exceed three for known examples. The article assumes that $P(\lambda)=\lambda^2$, 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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关键词
characteristic lie algebras,three-dimensional
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