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The Analytical Solutions of the Stochastic mKdV Equation via the Mapping Method

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  • Wael W. Mohammed

    (Department of Mathematics, Collage of Science, University of Ha’il, Ha’il 2440, Saudi Arabia
    Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt)

  • Farah M. Al-Askar

    (Department of Mathematical Science, Collage of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia)

  • Clemente Cesarano

    (Section of Mathematics, International Telematic University Uninettuno, Corso, Vittorio, Emanuele II 39, 00186 Roma, Italy)

Abstract

Here, we analyze the (2+1)-dimensional stochastic modified Kordeweg–de Vries (SmKdV) equation perturbed by multiplicative white noise in the Stratonovich sense. We apply the mapping method to obtain new trigonometric, elliptic, and rational stochastic fractional solutions. Because of the importance of the KdV equation in characterizing the behavior of waves in shallow water, the obtained solutions are beneficial in interpreting certain fascinating physical phenomena. We plot our figures in MATLAB and show several 3D and 2D graphical representations to show how the multiplicative white noise affects the solutions of the SmKdV. We show that the white noise around zero stabilizes SmKdV solutions.

Suggested Citation

  • Wael W. Mohammed & Farah M. Al-Askar & Clemente Cesarano, 2022. "The Analytical Solutions of the Stochastic mKdV Equation via the Mapping Method," Mathematics, MDPI, vol. 10(22), pages 1-9, November.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:22:p:4212-:d:969724
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    References listed on IDEAS

    as
    1. Wael W. Mohammed & Mohammed Alshammari & Clemente Cesarano & Sultan Albadrani & M. El-Morshedy, 2022. "Brownian Motion Effects on the Stabilization of Stochastic Solutions to Fractional Diffusion Equations with Polynomials," Mathematics, MDPI, vol. 10(9), pages 1-9, April.
    2. Younis, Muhammad & Rizvi, Syed Tahir Raza & Ali, Safdar, 2015. "Analytical and soliton solutions: Nonlinear model of nanobioelectronics transmission lines," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 994-1002.
    3. Alvaro H. Salas, 2012. "Solving Nonlinear Partial Differential Equations by the sn-ns Method," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-25, April.
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    Cited by:

    1. Wael W. Mohammed & Farah M. Al-Askar & Clemente Cesarano, 2023. "On the Dynamical Behavior of Solitary Waves for Coupled Stochastic Korteweg–De Vries Equations," Mathematics, MDPI, vol. 11(16), pages 1-17, August.
    2. Wael W. Mohammed & Farah M. Al-Askar & Clemente Cesarano & Mahmoud El-Morshedy, 2023. "Solitary Wave Solution of a Generalized Fractional–Stochastic Nonlinear Wave Equation for a Liquid with Gas Bubbles," Mathematics, MDPI, vol. 11(7), pages 1-14, April.
    3. Wael W. Mohammed & Farah M. Al-Askar & Clemente Cesarano & M. El-Morshedy, 2023. "Solitary Wave Solutions of the Fractional-Stochastic Quantum Zakharov–Kuznetsov Equation Arises in Quantum Magneto Plasma," Mathematics, MDPI, vol. 11(2), pages 1-14, January.
    4. Wael W. Mohammed & Farah M. Al-Askar & Clemente Cesarano & M. El-Morshedy, 2023. "On the Dynamics of Solitary Waves to a (3+1)-Dimensional Stochastic Boiti–Leon–Manna–Pempinelli Model in Incompressible Fluid," Mathematics, MDPI, vol. 11(10), pages 1-9, May.
    5. Wael W. Mohammed & Farah M. Al-Askar & Clemente Cesarano & Elkhateeb S. Aly, 2023. "The Soliton Solutions of the Stochastic Shallow Water Wave Equations in the Sense of Beta-Derivative," Mathematics, MDPI, vol. 11(6), pages 1-11, March.

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