Abstract:
Self-similar reductions of the Sawada – Kotera and Kupershmidt equations are studied. Results of Painlevé's test for these equations are given. Lax pairs for solving the Cauchy problems to these nonlinear ordinary differential equations are found. Special solutions of the Sawada – Kotera and Kupershmidt equations expressed via the first Painlevé equation are presented. Exact solutions of the Sawada – Kotera and Kupershmidt equations by means of general solution for the first member of K2 hierarchy are given. Special polynomials for expressions of rational solutions for the equations considered are introduced. The differentialdifference equations for finding special polynomials corresponding to the Sawada – Kotera and Kupershmidt equations are found. Nonlinear differential equations of sixth order for special polynomials associated with the Sawada – Kotera and Kupershmidt equations are obtained. Lax pairs for nonlinear differential equations with special polynomials are presented. Rational solutions of the self-similar reductions for the Sawada – Kotera and Kupershmidt equations are given.
Citation:
Nikolay A. Kudryashov, “Lax Pairs and Special Polynomials Associated with Self-similar Reductions of Sawada – Kotera and Kupershmidt Equations”, Regul. Chaotic Dyn., 25:1 (2020), 59–77
\Bibitem{Kud20}
\by Nikolay A. Kudryashov
\paper Lax Pairs and Special Polynomials Associated with Self-similar Reductions of Sawada – Kotera and Kupershmidt Equations
\jour Regul. Chaotic Dyn.
\yr 2020
\vol 25
\issue 1
\pages 59--77
\mathnet{http://mi.mathnet.ru/rcd1050}
\crossref{https://doi.org/10.1134/S1560354720010074}
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https://www.mathnet.ru/eng/rcd1050
https://www.mathnet.ru/eng/rcd/v25/i1/p59
This publication is cited in the following 5 articles:
O. González-Gaxiola, A. León-Ramírez, G. Chacón-Acosta, “Application of the Kudryashov Method for Finding
Exact Solutions of the Schamel – Kawahara Equation”, Rus. J. Nonlin. Dyn., 18:2 (2022), 203–215
Gangwei Wang, Li Li, Qi Wang, Juan Geng, “New Explicit Solutions of the Extended Double (2+1)-Dimensional Sine-Gorden Equation and Its Time Fractional Form”, Fractal Fract, 6:3 (2022), 166
Nikolay A. Kudryashov, “Lax Pairs and Rational Solutions of Similarity Reductions for
Kupershmidt and Sawada – Kotera Hierarchies”, Regul. Chaotic Dyn., 26:3 (2021), 271–292
Nikolay A. Kudryashov, “Rational Solutions of Equations Associated with the Second Painlevé Equation”, Regul. Chaotic Dyn., 25:3 (2020), 273–280
Oswaldo González-Gaxiola, Anjan Biswas, Mir Asma, Abdullah Kamis Alzahrani, “Optical Dromions and Domain Walls with the Kundu – Mukherjee – Naskar Equation by the Laplace – Adomian Decomposition Scheme”, Regul. Chaotic Dyn., 25:4 (2020), 338–348
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