New stochastic solitary solutions for the modified Korteweg-de Vries equation with stochastic term/random variable coefficients

Wael W. Mohammed; Farah M. Al-Askar; Wael W. Mohammed; Farah M. Al-Askar

doi:10.3934/math.2024995

AIMS Mathematics

2024, Volume 9, Issue 8: 20467-20481. doi: 10.3934/math.2024995

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New stochastic solitary solutions for the modified Korteweg-de Vries equation with stochastic term/random variable coefficients

Wael W. Mohammed ^{1,2
,
,},
Farah M. Al-Askar ³

1.
Department of Mathematics, College of Science, University of Ha'il, Ha'il 2440, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
3.
Department of Mathematical Science, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia

Received: 11 May 2024 Revised: 31 May 2024 Accepted: 11 June 2024 Published: 24 June 2024
MSC : 60H10, 60H15, 35Q51, 35A20, 83C15

In this research, we are examining the stochastic modified Korteweg-de Vries (SMKdV) equation forced in the Itô sense by multiplicative noise. We use an appropriate transformation to convert the SMKdV equation to another MKdV equation with random variable coefficients (MKdV-RVCs). We use the generalizing Riccati equation mapping and Jacobi elliptic functions methods in order to acquire new trigonometric, hyperbolic, and rational solutions for MKdV-RVCs. After that, we can get the solutions to the SMKdV equation. To our knowledge, this is the first time we have assumed that the solution of the wave equation for the SMKdV equation is stochastic, since all earlier research assumed that it was deterministic. Furthermore, we provide different graphic representations to show the influence of multiplicative noise on the exact solutions of the SMKdV equation.
- modified KdV equation,
- random variable coefficients,
- stability by noise,
- Jacobi elliptic functions method
Citation: Wael W. Mohammed, Farah M. Al-Askar. New stochastic solitary solutions for the modified Korteweg-de Vries equation with stochastic term/random variable coefficients[J]. AIMS Mathematics, 2024, 9(8): 20467-20481. doi: 10.3934/math.2024995

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Abstract

In this research, we are examining the stochastic modified Korteweg-de Vries (SMKdV) equation forced in the Itô sense by multiplicative noise. We use an appropriate transformation to convert the SMKdV equation to another MKdV equation with random variable coefficients (MKdV-RVCs). We use the generalizing Riccati equation mapping and Jacobi elliptic functions methods in order to acquire new trigonometric, hyperbolic, and rational solutions for MKdV-RVCs. After that, we can get the solutions to the SMKdV equation. To our knowledge, this is the first time we have assumed that the solution of the wave equation for the SMKdV equation is stochastic, since all earlier research assumed that it was deterministic. Furthermore, we provide different graphic representations to show the influence of multiplicative noise on the exact solutions of the SMKdV equation.

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