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Exact solutions of the equation for surface waves in a convecting fluid

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  • Kudryashov, Nikolay A.

Abstract

A method for finding exact solutions and the first integrals is presented. The basic idea of the method is to use the value of the Fuchs index that appears in the Painlevé test to construct the auxiliary equation for finding the first integrals and exact solutions of nonlinear differential equations. It allows us to obtain the first integrals and new exact solutions of some nonlinear ordinary differential equations. The main feature of the method is that we do not assign a solution function at the beginning, we find this function during calculations. This approach is conceptually equivalent to the third step of the Painlevé test and sometimes allows us to change this step. Our approach generalizes a number of other methods for finding exact solutions of nonlinear differential equations. We demonstrate a method for finding the traveling wave solutions and the first integrals of the well-known nonlinear evolution equation for description of surface waves in a convecting liquid. The general solution of this equation at some conditions on parameters and new traveling wave solutions of the fourth-order equation are found.

Suggested Citation

  • Kudryashov, Nikolay A., 2019. "Exact solutions of the equation for surface waves in a convecting fluid," Applied Mathematics and Computation, Elsevier, vol. 344, pages 97-106.
    • Handle: RePEc:eee:apmaco:v:344-345:y:2019:i::p:97-106
    DOI: 10.1016/j.amc.2018.10.005
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    1. Kudryashov, Nikolai A., 2005. "Simplest equation method to look for exact solutions of nonlinear differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 24(5), pages 1217-1231.
    2. Kudryashov, Nikolai A., 2005. "Fuchs indices and the first integrals of nonlinear differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 26(2), pages 591-603.
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    Cited by:

    1. Kudryashov, Nikolay A. & Kutukov, Aleksandr A. & Biswas, Anjan & Zhou, Qin & Yıldırım, Yakup & Alshomrani, Ali Saleh, 2023. "Optical solitons for the concatenation model: Power-law nonlinearity," Chaos, Solitons & Fractals, Elsevier, vol. 177(C).
    2. Liu, Hong-Zhun, 2022. "A modification to the first integral method and its applications," Applied Mathematics and Computation, Elsevier, vol. 419(C).
    3. Nikolay A. Kudryashov, 2021. "Implicit Solitary Waves for One of the Generalized Nonlinear Schrödinger Equations," Mathematics, MDPI, vol. 9(23), pages 1-9, November.
    4. Kudryashov, Nikolay A., 2020. "Highly dispersive solitary wave solutions of perturbed nonlinear Schrödinger equations," Applied Mathematics and Computation, Elsevier, vol. 371(C).
    5. Guan, Xue & Liu, Wenjun & Zhou, Qin & Biswas, Anjan, 2020. "Some lump solutions for a generalized (3+1)-dimensional Kadomtsev–Petviashvili equation," Applied Mathematics and Computation, Elsevier, vol. 366(C).
    6. Nikolay A. Kudryashov & Sofia F. Lavrova, 2023. "Painlevé Test, Phase Plane Analysis and Analytical Solutions of the Chavy–Waddy–Kolokolnikov Model for the Description of Bacterial Colonies," Mathematics, MDPI, vol. 11(14), pages 1-17, July.
    7. Kudryashov, Nikolay A., 2019. "Lax pair and first integrals of the traveling wave reduction for the KdV hierarchy," Applied Mathematics and Computation, Elsevier, vol. 350(C), pages 323-330.
    8. Kudryashov, Nikolay A., 2020. "First integrals and general solution of the complex Ginzburg-Landau equation," Applied Mathematics and Computation, Elsevier, vol. 386(C).
    9. Kudryashov, Nikolay A., 2020. "Optical solitons of model with integrable equation for wave packet envelope," Chaos, Solitons & Fractals, Elsevier, vol. 141(C).

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