Analyzing wave dynamics with noise effects: Exploring the stochastic Zhiber–Shabat model in engineering and mathematical physics

Jan Muhammad; Ali H. Tedjani; Usman Younas; Jan Muhammad; Ali H. Tedjani; Usman Younas

doi:10.3934/math.2026351

AIMS Mathematics

2026, Volume 11, Issue 3: 8521-8545. doi: 10.3934/math.2026351

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Analyzing wave dynamics with noise effects: Exploring the stochastic Zhiber–Shabat model in engineering and mathematical physics

1.
Department of Mathematics, Shanghai University, Shanghai 200444, China
2.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11564, Saudi Arabia

Received: 01 February 2026 Revised: 17 March 2026 Accepted: 23 March 2026 Published: 30 March 2026
MSC : 35A20, 35B10, 35C08, 60H15

This paper is concerned with analytical solutions of the stochastic Zhiber–Shabat equation with a multiplicative noise term. This model is said to be an excellent tool to investigate the behavior of integrable dynamics under uncertainty. Applications exist in many areas, such as optical communications, fluid mechanics, propagation of waves through complex media, and areas that interface between mathematics, physics, and data science. We examine solitary wave solutions of the proposed model by the modified generalized exponential rational function method, generalized Arnous method, and the modified F-expansion method. The suggested methodologies suggest different soliton solutions, which are dark, bright, exponential, bright-dark, periodic, and mixed-form solutions. A range of graphs with noise term effects is used to present the behavior of the solutions with respect to the various parametric values. This paper offers a new understanding phenomenon of nonlinear waves as it measures the effectiveness of contemporary mathematical tools and explains the peculiarities of the system dynamics.
- nonlinear dynamics,
- noise effect,
- solitons,
- mathematical methods,
- stochastic Zhiber–Shabat model
Citation: Jan Muhammad, Ali H. Tedjani, Usman Younas. Analyzing wave dynamics with noise effects: Exploring the stochastic Zhiber–Shabat model in engineering and mathematical physics[J]. AIMS Mathematics, 2026, 11(3): 8521-8545. doi: 10.3934/math.2026351

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Abstract

This paper is concerned with analytical solutions of the stochastic Zhiber–Shabat equation with a multiplicative noise term. This model is said to be an excellent tool to investigate the behavior of integrable dynamics under uncertainty. Applications exist in many areas, such as optical communications, fluid mechanics, propagation of waves through complex media, and areas that interface between mathematics, physics, and data science. We examine solitary wave solutions of the proposed model by the modified generalized exponential rational function method, generalized Arnous method, and the modified F-expansion method. The suggested methodologies suggest different soliton solutions, which are dark, bright, exponential, bright-dark, periodic, and mixed-form solutions. A range of graphs with noise term effects is used to present the behavior of the solutions with respect to the various parametric values. This paper offers a new understanding phenomenon of nonlinear waves as it measures the effectiveness of contemporary mathematical tools and explains the peculiarities of the system dynamics.

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