Constructing abundant exact stochastic solutions to the Newell-Whitehead-Segel equation with multiplicative advection noise

Abeer H. Alblowy; Wael W. Mohammed; Hessa W. Alshammari; Elsayed M. Elsayed; Abeer H. Alblowy; Wael W. Mohammed; Hessa W. Alshammari; Elsayed M. Elsayed

doi:10.3934/era.2026043

Electronic Research Archive

2026, Volume 34, Issue 2: 922-937. doi: 10.3934/era.2026043

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Constructing abundant exact stochastic solutions to the Newell-Whitehead-Segel equation with multiplicative advection noise

1.
Department of Mathematics, College of Science, University of Ha'il, Ha'il 2440, Saudi Arabia
2.
Mathematics Department, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
3.
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Received: 03 December 2025 Revised: 10 January 2026 Accepted: 24 January 2026 Published: 28 January 2026

In this work, we consider the stochastic Newell-Whitehead-Segel (SNWS) equation, which is forced in the Itô sense by multiplicative advection noise. We demonstrate that the solution of the SNWS equation can be discovered by solving certain deterministic counterparts of the Newell-Whitehead-Segel equation with an extra diffusion term (DNWS), and then the results are combined with a solution for stochastic ordinary differential equations. We use two techniques to solve the DNWS equation: the $ (\frac{G^{\prime }}{G}) $-expansion method and the $ \exp (-\psi (\eta)) $ -expansion method. The purpose of studying the Newell-Whitehead-Segel equation with multiplicative noise is to explore how stochastic fluctuations influence the stability and evolution of patterns near a bifurcation. Incorporating noise provides a more realistic description of physical systems and reveals impacts that are omitted from the deterministic model, such as changing thresholds, different pattern structures, and increased variability. Therefore, we use a MATLAB tool to exhibit numerous 3D and 2D graphs that show how multiplicative advection noise impacts the solutions of the SNWS equation.
- noise strength,
- simulation,
- stochastic solutions,
- analytical methods,
- mathematical model,
- exact solutions
Citation: Abeer H. Alblowy, Wael W. Mohammed, Hessa W. Alshammari, Elsayed M. Elsayed. Constructing abundant exact stochastic solutions to the Newell-Whitehead-Segel equation with multiplicative advection noise[J]. Electronic Research Archive, 2026, 34(2): 922-937. doi: 10.3934/era.2026043

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Abstract

In this work, we consider the stochastic Newell-Whitehead-Segel (SNWS) equation, which is forced in the Itô sense by multiplicative advection noise. We demonstrate that the solution of the SNWS equation can be discovered by solving certain deterministic counterparts of the Newell-Whitehead-Segel equation with an extra diffusion term (DNWS), and then the results are combined with a solution for stochastic ordinary differential equations. We use two techniques to solve the DNWS equation: the $ (\frac{G^{\prime }}{G}) $-expansion method and the $ \exp (-\psi (\eta)) $ -expansion method. The purpose of studying the Newell-Whitehead-Segel equation with multiplicative noise is to explore how stochastic fluctuations influence the stability and evolution of patterns near a bifurcation. Incorporating noise provides a more realistic description of physical systems and reveals impacts that are omitted from the deterministic model, such as changing thresholds, different pattern structures, and increased variability. Therefore, we use a MATLAB tool to exhibit numerous 3D and 2D graphs that show how multiplicative advection noise impacts the solutions of the SNWS equation.

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