On the~problem of classifying integrable chains with three independent variables
Teoretičeskaâ i matematičeskaâ fizika, Tome 215 (2023) no. 2, pp. 242-268
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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 $\deg P(\lambda)=2$.
Keywords:
three-dimensional chains, characteristic algebras, Darboux integrability, characteristic integrals, integrable reductions.
@article{TMF_2023_215_2_a6,
author = {M. N. Kuznetsova and I. T. Habibullin and A. R. Khakimova},
title = {On the~problem of classifying integrable chains with three independent variables},
journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
pages = {242--268},
publisher = {mathdoc},
volume = {215},
number = {2},
year = {2023},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TMF_2023_215_2_a6/}
}
TY - JOUR AU - M. N. Kuznetsova AU - I. T. Habibullin AU - A. R. Khakimova TI - On the~problem of classifying integrable chains with three independent variables JO - Teoretičeskaâ i matematičeskaâ fizika PY - 2023 SP - 242 EP - 268 VL - 215 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TMF_2023_215_2_a6/ LA - ru ID - TMF_2023_215_2_a6 ER -
%0 Journal Article %A M. N. Kuznetsova %A I. T. Habibullin %A A. R. Khakimova %T On the~problem of classifying integrable chains with three independent variables %J Teoretičeskaâ i matematičeskaâ fizika %D 2023 %P 242-268 %V 215 %N 2 %I mathdoc %U http://geodesic.mathdoc.fr/item/TMF_2023_215_2_a6/ %G ru %F TMF_2023_215_2_a6
M. N. Kuznetsova; I. T. Habibullin; A. R. Khakimova. On the~problem of classifying integrable chains with three independent variables. Teoretičeskaâ i matematičeskaâ fizika, Tome 215 (2023) no. 2, pp. 242-268. http://geodesic.mathdoc.fr/item/TMF_2023_215_2_a6/