The influence of stochastic process on some new solutions for the long-short-wave interaction system

Mahmoud A. E. Abdelrahman; Yousef F. Alharbi; Mahmoud A. E. Abdelrahman; Yousef F. Alharbi

doi:10.3934/math.20251083

AIMS Mathematics

2025, Volume 10, Issue 10: 24431-24445. doi: 10.3934/math.20251083

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The influence of stochastic process on some new solutions for the long-short-wave interaction system

Mahmoud A. E. Abdelrahman ^{1,2
,
,},
Yousef F. Alharbi ¹

1.
Department of Mathematics, College of Science, Taibah University, Madinah, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt

Received: 15 June 2025 Revised: 24 September 2025 Accepted: 28 September 2025 Published: 24 October 2025
MSC : 35C05, 60G07, 35R10, 35R60

The nonlinear long-short wave interaction system serves as a fundamental nonlinear model that characterizes the resonant interactions occurring between high-frequency (short) waves and low-frequency (long) waves. This system is especially useful for comprehending energy transfer, wave modulation, and the development of localized structures such as solitons or breathers. This paper proposes innovative stochastic solutions for the nonlinear long-short wave interaction model within the context of the Brownian motion process. This stochastic process takes into consideration the intrinsic randomness and variations found in real-world systems, including nonlinear optical fibers, Bose-Einstein condensates, fluid dynamics, and plasma environments. We derive stochastic traveling wave solutions using an extended tanh function approach and examine the resulting stochastic dynamics concerning amplitude variation, phase shifts, and noise-induced modulation. We consider the impact of noise intensity on the stability and coherence of wave structures. Our findings suggest that Brownian forcing can fulfill two roles; either aiding in the preservation of localized structures or leading to their collapse and dispersion, depending on the system parameters and initial conditions. To depict the behavior of the designated stochastic solutions, various wave profiles were generated utilizing the MATLAB software. Finally, the proposed method holds the promise of being adapted for various other practical models.
- Brownian motion process,
- long-short-wave,
- soliton solutions,
- noise intensity
Citation: Mahmoud A. E. Abdelrahman, Yousef F. Alharbi. The influence of stochastic process on some new solutions for the long-short-wave interaction system[J]. AIMS Mathematics, 2025, 10(10): 24431-24445. doi: 10.3934/math.20251083

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Abstract

The nonlinear long-short wave interaction system serves as a fundamental nonlinear model that characterizes the resonant interactions occurring between high-frequency (short) waves and low-frequency (long) waves. This system is especially useful for comprehending energy transfer, wave modulation, and the development of localized structures such as solitons or breathers. This paper proposes innovative stochastic solutions for the nonlinear long-short wave interaction model within the context of the Brownian motion process. This stochastic process takes into consideration the intrinsic randomness and variations found in real-world systems, including nonlinear optical fibers, Bose-Einstein condensates, fluid dynamics, and plasma environments. We derive stochastic traveling wave solutions using an extended tanh function approach and examine the resulting stochastic dynamics concerning amplitude variation, phase shifts, and noise-induced modulation. We consider the impact of noise intensity on the stability and coherence of wave structures. Our findings suggest that Brownian forcing can fulfill two roles; either aiding in the preservation of localized structures or leading to their collapse and dispersion, depending on the system parameters and initial conditions. To depict the behavior of the designated stochastic solutions, various wave profiles were generated utilizing the MATLAB software. Finally, the proposed method holds the promise of being adapted for various other practical models.

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