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

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Abstract

We continue describing integrable nonlinear chains of the form \(u^j_{n+1,x}=u^j_{n,x}+f(u^{j+1}_{n},u^{j}_n,u^j_{n+1 },u^{j-1}_{n+1})\) with three independent variables on the basis of the existence of a hierarchy of Darboux-integrable reductions. The classification algorithm is based on the well-known fact that characteristic algebras of Darboux-integrable systems have a finite dimension. We use a characteristic algebra in the \(x\)-direction, whose structure for a given class of models is defined by some polynomial \(P(\lambda)\) of degree not exceeding \(3\) in the known examples. We assume that \(P(\lambda)=\lambda^2\), the classification problem in that case reduces to finding eight unknown functions of a single variable. We obtain a rather narrow class of candidates for the integrability.

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Notes

  1. We use the scheme proposed previously in [29].

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Funding

The work was supported by the Russian Science Foundation (grant No. 21-11-00006, https://rscf.ru/en/project/21-11-00006/).

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Authors and Affiliations

  1. Institute of Mathematics with Computing Center of the Ufa Federal Research Center, Russian Academy of Sciences, Ufa, Russia

    I. T. Habibullin & A. R. Khakimova

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  1. I. T. Habibullin
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  2. A. R. Khakimova
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Correspondence to I. T. Habibullin.

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Translated from Teoreticheskaya i Matematicheskaya Fizika, 2023, Vol. 217, pp. 142–178 https://doi.org/10.4213/tmf10513.

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Habibullin, I.T., Khakimova, A.R. On the classification of nonlinear integrable three-dimensional chains via characteristic Lie algebras. Theor Math Phys 217, 1541–1573 (2023). https://doi.org/10.1134/S0040577923100094

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  • DOI: https://doi.org/10.1134/S0040577923100094

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