Classification of semidiscrete equations of hyperbolic type. The case of third-order symmetries. (English. Russian original) Zbl 1536.37064
Theor. Math. Phys. 217, No. 2, 1767-1776 (2023); translation from Teor. Mat. Fiz. 217, No. 2, 404-415 (2023).
Summary: We classify semidiscrete equations of hyperbolic type. We study the class of equations of the form
\[\frac{du_{n+1}}{dx}=f\left(\frac{du_n}{dx},u_{n+1},u_n\right),\] 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.
MSC:
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 37K60 | Lattice dynamics; integrable lattice equations |
| 39A36 | Integrable difference and lattice equations; integrability tests |
| 34K04 | Symmetries, invariants of functional-differential equations |
| 34K31 | Lattice functional-differential equations |
Keywords:
integrability; generalized symmetry; classification; semidiscrete equation; hyperbolic typeReferences:
| [1] | Veselov, A. P.; Shabat, A. B., Dressing chains and spectral theory of the Schrödinger operator, Funct. Anal. Appl., 27, 81-96 (1993) · Zbl 0813.35099 · doi:10.1007/BF01085979 |
| [2] | Yamilov, R., Symmetries as integrability criteria for differential difference equations, J. Phys. A: Math. Gen., 39, R541-R623 (2006) · Zbl 1105.35136 · doi:10.1088/0305-4470/39/45/R01 |
| [3] | Yamilov, R. I., Classification of discrete evolution equations, Upsekhi Mat. Nauk, 38, 155-156 (1983) |
| [4] | Yamilov, R. I., Invertible changes of variables generated by Bäcklund transformations, Theoret. and Math. Phys., 85, 1269-1275 (1990) · Zbl 0726.35117 · doi:10.1007/BF01018403 |
| [5] | Garifullin, R. N.; Habibullin, I. T., Generalized symmetries and integrability conditions for hyperbolic type semi-discrete equations, J. Phys. A: Math. Theor., 54 (2021) · Zbl 1519.37096 · doi:10.1088/1751-8121/abf3ea |
| [6] | Meshkov, A. G.; Sokolov, V. V., Integrable evolution equations with a constant separant, Ufa Math. J., 4, 104-152 (2012) · Zbl 1299.35159 |
| [7] | Svinolupov, S. I.; Sokolov, V. V., Evolution equations with nontrivial conservative laws, Funct. Anal. Appl., 16, 317-319 (1982) · Zbl 0508.35072 · doi:10.1007/BF01077866 |
| [8] | Garifullin, R. N.; Yamilov, R. I., Generalized symmetry classification of discrete equations of a class depending on twelve parameters, J. Phys. A: Math. Theor., 45 (2012) · Zbl 1257.39007 · doi:10.1088/1751-8113/45/34/345205 |
| [9] | Garifullin, R. N.; Mikhailov, A. V.; Yamilov, R. I., Discrete equation on a square lattice with a nonstandard structure of generalized symmetries, Theoret. and Math. Phys., 180, 765-780 (2014) · Zbl 1311.39007 · doi:10.1007/s11232-014-0178-6 |
| [10] | Garifullin, R. N.; Yamilov, R. I., An unusual series of autonomous discrete integrable equations on a square lattice, Theoret. and Math. Phys., 200, 966-984 (2019) · Zbl 1430.39005 · doi:10.1134/S0040577919070031 |
| [11] | Adler, V. E., Bäcklund transformation for the Krichever-Novikov equation, Internat. Math. Res. Notices, 1998, 1-4 (1998) · Zbl 0895.35089 · doi:10.1155/S1073792898000014 |
| [12] | Startsev, S. Ya., Darboux integrable differential-difference equations admitting a first-order integral, Ufa Math. J., 4, 159-173 (2012) |
| [13] | Yamilov, R. I., On the construction of Miura type transformations by others of this kind, Phys. Lett. A, 173, 53-57 (1993) · doi:10.1016/0375-9601(93)90086-F |
| [14] | Startsev, S. Ya., Non-point invertible transformations and integrability of partial difference equations, SIGMA, 10 (2014) · Zbl 1297.39013 |
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