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Fractal Fract
Volume 9
Issue 11
10.3390/fractalfract9110728
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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

Nonlinear Stochastic Wave Behavior: Soliton Solutions and Energy Analysis of Kairat-II and Kairat-X Systems

by
Syed T. R. Rizvi
1,
Lotfi Jlali
2,
Iqra Anjum
1,
Husnain Abad
1,
Emad Solouma
2 and
Aly R. Seadawy
3,*
1
Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore 54000, Pakistan
2
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi Arabia
3
Mathematics Department, Faculty of Science, Taibah University, Madinah 41411, Saudi Arabia
*
Author to whom correspondence should be addressed.
Fractal Fract. 2025, 9(11), 728; https://doi.org/10.3390/fractalfract9110728
Submission received: 27 September 2025 / Revised: 2 November 2025 / Accepted: 7 November 2025 / Published: 11 November 2025
(This article belongs to the Special Issue Analysis and Numerical Computations of Nonlinear Fractional and Classical Differential Equations)
Versions Notes

Abstract

We study stochastic variants of the Kairat-II and Kairat-X equations in (3 + 1) dimensions, two canonical models in soliton theory. Random fluctuations are incorporated through a Wiener process, yielding a multiplicative stochastic embedding of the wave fields. By combining the enhanced direct algebraic technique with the new projective Riccati equation approach, we obtain closed-form stochastic soliton solutions and analyze how noise modulates their amplitude and localization. The solutions are illustrated with consistent 3D surface plots (mean field vs. sample paths) and 2D time traces to highlight wave geometry and variability. In addition, we employ the energy balance approach to separate kinetic and potential contributions and to verify an energy balance relation for the derived solutions, thereby clarifying their physical plausibility and stability under noise. The results provide exact, easily verifiable benchmarks for stochastic nonlinear wave models and a practical template for incorporating randomness into nonlinear dispersive systems.
Keywords: stochastic differential equation; soliton solutions; Kairat-II and Kairat-X equations; energy balance approach stochastic differential equation; soliton solutions; Kairat-II and Kairat-X equations; energy balance approach

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MDPI and ACS Style

Rizvi, S.T.R.; Jlali, L.; Anjum, I.; Abad, H.; Solouma, E.; Seadawy, A.R. Nonlinear Stochastic Wave Behavior: Soliton Solutions and Energy Analysis of Kairat-II and Kairat-X Systems. Fractal Fract. 2025, 9, 728. https://doi.org/10.3390/fractalfract9110728

AMA Style

Rizvi STR, Jlali L, Anjum I, Abad H, Solouma E, Seadawy AR. Nonlinear Stochastic Wave Behavior: Soliton Solutions and Energy Analysis of Kairat-II and Kairat-X Systems. Fractal and Fractional. 2025; 9(11):728. https://doi.org/10.3390/fractalfract9110728

Chicago/Turabian Style

Rizvi, Syed T. R., Lotfi Jlali, Iqra Anjum, Husnain Abad, Emad Solouma, and Aly R. Seadawy. 2025. "Nonlinear Stochastic Wave Behavior: Soliton Solutions and Energy Analysis of Kairat-II and Kairat-X Systems" Fractal and Fractional 9, no. 11: 728. https://doi.org/10.3390/fractalfract9110728

APA Style

Rizvi, S. T. R., Jlali, L., Anjum, I., Abad, H., Solouma, E., & Seadawy, A. R. (2025). Nonlinear Stochastic Wave Behavior: Soliton Solutions and Energy Analysis of Kairat-II and Kairat-X Systems. Fractal and Fractional, 9(11), 728. https://doi.org/10.3390/fractalfract9110728

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