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Painlevé Test, Phase Plane Analysis and Analytical Solutions of the Chavy–Waddy–Kolokolnikov Model for the Description of Bacterial Colonies

Author

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  • Nikolay A. Kudryashov

    (Moscow Engineering Physics Institute, National Research Nuclear University MEPhI, 31 Kashirskoe Shosse, 115409 Moscow, Russia)

  • Sofia F. Lavrova

    (Moscow Engineering Physics Institute, National Research Nuclear University MEPhI, 31 Kashirskoe Shosse, 115409 Moscow, Russia)

Abstract

The Chavy–Waddy–Kolokolnikov model for the description of bacterial colonies is considered. In order to establish if the mathematical model is integrable, the Painlevé test is conducted for the nonlinear ordinary differential equation which corresponds to the fourth-order partial differential equation. The restrictions on the mathematical model parameters for ordinary differential equations to pass the Painlevé test are obtained. It is determined that the method of the inverse scattering transform does not solve the Cauchy problem for the original mathematical model, since the corresponding nonlinear ordinary differential equation passes the Painlevé test only when its solution is stationary. In the case of the stationary solution, the first integral of the equation is obtained, which makes it possible to represent the general solution in the quadrature form. The stability of the stationary points of the investigated mathematical model is carried out and their classification is proposed. Periodic and solitary stationary solutions of the Chavy–Waddy–Kolokolnikov model are constructed for various parameter values. To build analytical solutions, the method of the simplest equations is also used. The solutions, obtained in the form of a truncated expansion in powers of the logistic function, are represented as a closed formula using the formula for the Newton binomial.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:14:p:3203-:d:1199328
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    References listed on IDEAS

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    1. Arzu Akbulut & Gaetano Luciano, 2023. "Obtaining the Soliton Type Solutions of the Conformable Time-Fractional Complex Ginzburg–Landau Equation with Kerr Law Nonlinearity by Using Two Kinds of Kudryashov Methods," Journal of Mathematics, Hindawi, vol. 2023, pages 1-6, February.
    2. Nikolay A. Kudryashov, 2022. "Optical Solitons of the Generalized Nonlinear Schrödinger Equation with Kerr Nonlinearity and Dispersion of Unrestricted Order," Mathematics, MDPI, vol. 10(18), pages 1-9, September.
    3. 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.
    4. Arnous, Ahmed H. & Biswas, Anjan & Yıldırım, Yakup & Zhou, Qin & Liu, Wenjun & Alshomrani, Ali S. & Alshehri, Hashim M., 2022. "Cubic–quartic optical soliton perturbation with complex Ginzburg–Landau equation by the enhanced Kudryashov’s method," Chaos, Solitons & Fractals, Elsevier, vol. 155(C).
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    1. Nikolay A. Kudryashov & Sofia F. Lavrova & Daniil R. Nifontov, 2023. "Bifurcations of Phase Portraits, Exact Solutions and Conservation Laws of the Generalized Gerdjikov–Ivanov Model," Mathematics, MDPI, vol. 11(23), pages 1-20, November.

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