On the problem of classifying integrable chains with three independent variables

M. N. Kuznetsova   ; I. T. Habibullin   ; A. R. Khakimova

Teoretičeskaâ i matematičeskaâ fizika, Tome 215 (2023) no. 2, pp. 242-268

Résumé

We discuss a new method for the classification of integrable nonlinear chains with three independent variables using an example of chains in the form $u^j_{n+1,x}=u^j_{n,x}+f(u^{j+1}_{n},u^{j}_n,u^j_{n+1 },u^{j-1}_{n+1})$. This method is based on reductions having the form of systems of differential–difference Darboux-integrable equations. It is well known that the characteristic algebras of Darboux-integrable systems have a finite dimension. The structure of the characteristic algebra is defined by some polynomial $P(\lambda)$. The polynomial degree for the known integrable chains from the class under consideration equals $2$ or $3$. A partial classification is performed in the case $\deg P(\lambda)=2$.

Détail

Keywords: three-dimensional chains, characteristic algebras, Darboux integrability, characteristic integrals, integrable reductions.

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