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Analytic solutions of a microstructure PDE and the KdV and Kadomtsev–Petviashvili equations by invariant Painlevé analysis and generalized Hirota techniques

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  • Russo, Matthew
  • Choudhury, S. Roy

Abstract

Truncated Painlevé expansions, invariant Painlevé analysis, and generalized Hirota expansions are employed in combination to solve (‘partially reduce to quadrature’) the integrable KdV and KP equations, and a nonintegrable generalized microstructure (GMS) equation. Although the multisolitons of the KdV and KP equations are very well-known, the solutions obtained here for all the three NLPDEs are novel and non-trivial. The solutions obtained via invariant Painlevé analysis are all complicated rational functions, with arguments which themselves are confluent hypergeometric (KdV) or trigonometric (GMS) functions of various distinct non-traveling (KdV) and traveling wave variables. In some cases, this is slightly reminiscent of doubly-periodic elliptic function solutions when nonlinear ODE systems are reduced to quadratures. The solutions obtained by the use of recently-generalized Hirota-type expansions in the truncated Painlevé expansions are closer in functional form to conventional hyperbolic secant solutions, although with non-trivial traveling-wave arguments which are distinct for the three NLPDEs considered here.

Suggested Citation

  • Russo, Matthew & Choudhury, S. Roy, 2017. "Analytic solutions of a microstructure PDE and the KdV and Kadomtsev–Petviashvili equations by invariant Painlevé analysis and generalized Hirota techniques," Applied Mathematics and Computation, Elsevier, vol. 311(C), pages 228-239.
    • Handle: RePEc:eee:apmaco:v:311:y:2017:i:c:p:228-239
    DOI: 10.1016/j.amc.2017.01.055
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    Cited by:

    1. Verma, Pallavi & Kaur, Lakhveer, 2019. "Integrability, bilinearization and analytic study of new form of (3+1)-dimensional B-type Kadomstev–Petviashvili (BKP)- Boussinesq equation," Applied Mathematics and Computation, Elsevier, vol. 346(C), pages 879-886.
    2. Maria Santos Bruzón & Gaetana Gambino & Maria Luz Gandarias, 2021. "Generalized Camassa–Holm Equations: Symmetry, Conservation Laws and Regular Pulse and Front Solutions," Mathematics, MDPI, vol. 9(9), pages 1-20, April.
    3. Baoyong Guo & Huanhe Dong & Yong Fang, 2019. "Lump Solutions and Interaction Solutions for the Dimensionally Reduced Nonlinear Evolution Equation," Complexity, Hindawi, vol. 2019, pages 1-9, October.

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