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Lie symmetry, nonlocal symmetry analysis, and interaction of solutions of a \((2+1)\)-dimensional KdV-mKdV equation. (English. Russian original) Zbl 1467.35297

Theor. Math. Phys. 206, No. 2, 142-162 (2021); translation from Teor. Mat. Fiz. 206, No. 2, 164-185 (2021).
Summary: We use the method of Lie symmetry analysis to investigate the properties of a \((2+1)\)-dimensional KdV-mKdV equation. Using the Ibragimov method, which relies only on the existence of the commutator table, we construct an optimal system of one-dimensional subalgebras of the Lie algebra and study invariant solutions and similarity reductions by considering representatives of the optimal system. To analyze some nonlocal symmetry properties, we apply the truncated Painlevé expansion method and obtain two Bäcklund transformations that are not autotransformations and one auto-Bäcklund transformation. To localize the nonlocal symmetry and obtain a local Lie point symmetry, we introduce an expanded system. Using solutions of the corresponding Cauchy problems for Lie point symmetries, we prove a theorem on a finite symmetry transformation and find the \(n\) th Bäcklund transformation in terms of determinants. Based on one of the obtained Bäcklund transformations that are not autotransformations, we derive lump-type solutions. In addition, we prove the integrability of the equation by the consistent Riccati expansion method. We present explicit soliton-cnoidal wave solutions and investigate the dynamical characteristics of the obtained solutions using numerical analysis.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
35C08 Soliton solutions
17B80 Applications of Lie algebras and superalgebras to integrable systems
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