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Symbolic methods to construct exact solutions of nonlinear partial differential equations

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  • Hereman, Willy
  • Nuseir, Ameina

Abstract

Two straightforward methods for finding solitary-wave and soliton solutions are presented and applied to a variety of nonlinear partial differential equations. The first method is a simplied version of Hirota's method. It is shown to be an effective tool to explicitly construct. multi-soliton solutions of completely integrable evolution equations of fifth-order, including the Kaup-Kupershmidt equation for which the soliton solutions were not previously known. The second technique is the truncated Painlevé expansion method or singular manifold method. It is used to find closed-form solitary-wave solutions of the Fitzhugh-Nagumo equation with convection term, and an evolution equation due to Calogero. Since both methods are algorithmic, they can be implemented in the language of any symbolic manipulation program.

Suggested Citation

  • Hereman, Willy & Nuseir, Ameina, 1997. "Symbolic methods to construct exact solutions of nonlinear partial differential equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 43(1), pages 13-27.
    • Handle: RePEc:eee:matcom:v:43:y:1997:i:1:p:13-27
    DOI: 10.1016/S0378-4754(96)00053-5
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    Cited by:

    1. Zarea, Sana’a A., 2009. "The tanh method: A tool for solving some mathematical models," Chaos, Solitons & Fractals, Elsevier, vol. 41(2), pages 979-988.
    2. Al-Mdallal, Qasem M. & Syam, Muhammad I., 2007. "Sine–Cosine method for finding the soliton solutions of the generalized fifth-order nonlinear equation," Chaos, Solitons & Fractals, Elsevier, vol. 33(5), pages 1610-1617.
    3. Sartanpara, Parthkumar P. & Meher, Ramakanta, 2023. "Solution of generalised fuzzy fractional Kaup–Kupershmidt equation using a robust multi parametric approach and a novel transform," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 205(C), pages 939-969.
    4. Abdul-Majid Wazwaz & Ma’mon Abu Hammad & Ali O. Al-Ghamdi & Mansoor H. Alshehri & Samir A. El-Tantawy, 2023. "New (3+1)-Dimensional Kadomtsev–Petviashvili–Sawada– Kotera–Ramani Equation: Multiple-Soliton and Lump Solutions," Mathematics, MDPI, vol. 11(15), pages 1-11, August.
    5. Kudryashov, N.A. & Lavrova, S.F., 2021. "Dynamical features of the generalized Kuramoto-Sivashinsky equation," Chaos, Solitons & Fractals, Elsevier, vol. 142(C).
    6. Wazwaz, Abdul-Majid, 2015. "New (3+1)-dimensional nonlinear evolution equations with mKdV equation constituting its main part: Multiple soliton solutions," Chaos, Solitons & Fractals, Elsevier, vol. 76(C), pages 93-97.
    7. Biswas, Swapan & Ghosh, Uttam & Raut, Santanu, 2023. "Construction of fractional granular model and bright, dark, lump, breather types soliton solutions using Hirota bilinear method," Chaos, Solitons & Fractals, Elsevier, vol. 172(C).
    8. Khater, A.H. & Malfliet, W. & Kamel, E.S., 2004. "Travelling wave solutions of some classes of nonlinear evolution equations in (1+1) and higher dimensions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 64(2), pages 247-258.

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