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Classification of semidiscrete equations of hyperbolic type. The case of third-order symmetries

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Abstract

We classify semidiscrete equations of hyperbolic type. We study the class of equations of the form

$$\frac{du_{n+1}}{dx}=f\biggl(\frac{du_{n}}{dx},u_{n+1},u_{n}\biggr),$$

where the unknown function \(u_n(x)\) depends on one discrete (\(n\)) and one continuous (\(x\)) variables. The classification is based on the requirement that generalized symmetries exist in the discrete and continuous directions. We consider the case where the symmetries are of order \(3\) in both directions. As a result, a list of equations with the required conditions is obtained.

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Notes

  1. The terminology belongs to F. Calogero.

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Acknowledgments

The author is grateful to the anonymous referee for the critical comments and a valuable advice.

Funding

This research 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, Ufa Federal Research Center, Russian Academy of Science, Ufa, Russia

    R. N. Garifullin

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  1. R. N. Garifullin
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Correspondence to R. N. Garifullin.

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Additional information

Translated from Teoreticheskaya i Matematicheskaya Fizika, 2023, Vol. 217, pp. 404–415 https://doi.org/10.4213/tmf10512.

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Garifullin, R.N. Classification of semidiscrete equations of hyperbolic type. The case of third-order symmetries. Theor Math Phys 217, 1767–1776 (2023). https://doi.org/10.1134/S0040577923110119

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  • DOI: https://doi.org/10.1134/S0040577923110119

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