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Volume 13
Issue 13
10.3390/math13132193
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Article

Dynamical Behavior of Solitary Waves for the Space-Fractional Stochastic Regularized Long Wave Equation via Two Distinct Approaches

by
Muneerah Al Nuwairan
*,
Bashayr Almutairi
and
Anwar Aldhafeeri
Department of Mathematics and Statistics, College of Science, King Faisal University, P. O. Box 400, Al-Ahsa 31982, Saudi Arabia
*
Author to whom correspondence should be addressed.
Mathematics 2025, 13(13), 2193; https://doi.org/10.3390/math13132193
Submission received: 15 June 2025 / Revised: 1 July 2025 / Accepted: 3 July 2025 / Published: 4 July 2025
Versions Notes

Abstract

This study investigates the influence of multiplicative noise—modeled by a Wiener process—and spatial-fractional derivatives on the dynamics of the space-fractional stochastic Regularized Long Wave equation. By employing a complete discriminant polynomial system, we derive novel classes of fractional stochastic solutions that capture the complex interplay between stochasticity and nonlocality. Additionally, the variational principle, derived by He’s semi-inverse method, is utilized, yielding additional exact solutions that are bright solitons, bright-like solitons, kinky bright solitons, and periodic structures. Graphical analyses are presented to clarify how variations in the fractional order and noise intensity affect essential solution features, such as amplitude, width, and smoothness, offering deeper insight into the behavior of such nonlinear stochastic systems.
Keywords: stochastic fractional differential equations; variational principle; semi inverse method; complete discriminate system; long wave equation; modified Riemann–Liouville derivative stochastic fractional differential equations; variational principle; semi inverse method; complete discriminate system; long wave equation; modified Riemann–Liouville derivative

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MDPI and ACS Style

Al Nuwairan, M.; Almutairi, B.; Aldhafeeri, A. Dynamical Behavior of Solitary Waves for the Space-Fractional Stochastic Regularized Long Wave Equation via Two Distinct Approaches. Mathematics 2025, 13, 2193. https://doi.org/10.3390/math13132193

AMA Style

Al Nuwairan M, Almutairi B, Aldhafeeri A. Dynamical Behavior of Solitary Waves for the Space-Fractional Stochastic Regularized Long Wave Equation via Two Distinct Approaches. Mathematics. 2025; 13(13):2193. https://doi.org/10.3390/math13132193

Chicago/Turabian Style

Al Nuwairan, Muneerah, Bashayr Almutairi, and Anwar Aldhafeeri. 2025. "Dynamical Behavior of Solitary Waves for the Space-Fractional Stochastic Regularized Long Wave Equation via Two Distinct Approaches" Mathematics 13, no. 13: 2193. https://doi.org/10.3390/math13132193

APA Style

Al Nuwairan, M., Almutairi, B., & Aldhafeeri, A. (2025). Dynamical Behavior of Solitary Waves for the Space-Fractional Stochastic Regularized Long Wave Equation via Two Distinct Approaches. Mathematics, 13(13), 2193. https://doi.org/10.3390/math13132193

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