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:
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