Abstract
We study the symmetry and integrability of a Generalized Modified Camassa-Holm Equation (GMCH) of the form
We observe that for all increasing values of \(n\in {\mathbb {R}}\), \({\mathbb {R}}\) denotes the set of real number, the above equation gives a family of equations in which nonlinearity is rapidly increasing as n increases. However, this family has similar form of symmetries, a commutator table, an adjoint representation, and a one-dimensional optimal system. Interestingly, we show that the resultant second-order nonlinear ODE generated from the GMCH equation is linearizable because it possesses maximal symmetries. Finally, we conclude that the GMCH family passes the Painlevé Test since the resultant third-order nonlinear ordinary differential equation passes the Painlevé Test. This family does, in fact, have a similar form of leading order, resonances and truncated series of solution too.
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Acknowledgements
KK and ADD thank Prof. Stylianos Dimas, Sáo José dos Campos/SP, Brasil, for providing a new version of the SYM-Package. KK thank Late Prof. K.M.Tamizhmani for his support and tremendous academic guidance during his Ph.D.
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Communicated by V D Sharma.
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Krishnakumar, K., Devi, A.D., Srinivasan, V. et al. Optimal system, similarity solution and Painlevé test on generalized modified Camassa-Holm equation. Indian J Pure Appl Math 54, 547–557 (2023). https://doi.org/10.1007/s13226-022-00274-1
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DOI: https://doi.org/10.1007/s13226-022-00274-1