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Symmetries and integrability of the modified Camassa–Holm equation with an arbitrary parameter

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Abstract

We study the symmetry and integrability of a modified Camassa–Holm equation (MCH), with an arbitrary parameter k,  of the form

$$\begin{aligned} u_{t}+k(u-u_{xx})^2u_{x}-u_{xxt}+(u^{2}-{u_{x}}^2)(u_{x}-u_{xxx})=0. \end{aligned}$$

The commutator table and adjoint representation of the symmetries are presented to construct one-dimensional optimal system. By using the one-dimensional optimal system, we reduce the order or number of independent variables of the above equation and also we obtain interesting novel solutions for the reduced ordinary differential equations. Finally, we apply the Painlevé test to the resultant nonlinear ordinary differential equation and it is observed that the equation is integrable.

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Acknowledgements

ADD and KK thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of this article. They also would like to thank Prof. Stylianos Dimas, Sáo José dos Campos/SP, Brasil, for providing us a new version of the SYM-Package. PGLL thanks the National Research Foundation of South Africa for its continued support.

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Authors and Affiliations

  1. Department of Physics, Srinivasa Ramanujan Centre, SASTRA Deemed to be University, Kumbakonam, 612 001, India

    A Durga Devi

  2. Department of Mathematics, Srinivasa Ramanujan Centre, SASTRA Deemed to be University, Kumbakonam, 612 001, India

    K Krishnakumar

  3. Department of Mathematics, VIT-AP University, Amaravathi, 522 237, India

    R Sinuvasan

  4. Department of Mathematics, Durban University of Technology, P.O. Box 1334, Durban, 4000, Republic of South Africa

    P G L Leach

Authors
  1. A Durga Devi
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  2. K Krishnakumar
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  3. R Sinuvasan
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  4. P G L Leach
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Correspondence to K Krishnakumar.

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Devi, A.D., Krishnakumar, K., Sinuvasan, R. et al. Symmetries and integrability of the modified Camassa–Holm equation with an arbitrary parameter. Pramana - J Phys 95, 85 (2021). https://doi.org/10.1007/s12043-021-02103-2

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  • DOI: https://doi.org/10.1007/s12043-021-02103-2

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