In this study, we examined the impact of stochastic factors on the dynamics of nonlinear wave propagation by developing a damped modified Korteweg-de Vries (mKdV) equation incorporating multiplicative noise, mathematically characterized by a Wiener process. Standard deterministic models, although proficient in idealized contexts, frequently fail to accurately depict the complex behaviors elicited by the stochastic fluctuations intrinsic to physical systems. To mitigate this limitation, we utilized a hybrid analytical-numerical approach, employing the modified simple equation method to obtain a spectrum of precise soliton and solitary wave solutions. We ran complementary numerical simulations to find out how different levels of noise affect the time evolution and structural stability of these waveforms. The results showed that solitons keep their structure intact when there is not much noise. Nevertheless, as the noise level grows, the amplitude modulation and potential destabilization become more noticeable. Graphs like density maps and three-dimensional surface plots can be used to see how random changes make waves less predictable. These results demonstrated how crucial it is for nonlinear wave models to include random parts. Future research on complex noise profiles, multi-variable systems, and real-world validation approaches will benefit from this. A stochastic damped mKdV framework, which incorporates multiplicative noise with traditional deterministic or additive-noise analysis, is presented in this article. This paints a fuller picture of the way in which randomness evolves in response to actual system states.
Citation: Mohra Zayed, Hanadi M. AbdelSalam, Ibtisam Daqqa, S. Abdel-Khalek, E A-B Abdel-Salam, Gamal M. Ismail. Stochastic dynamics of solitary waves in the damped mKdV equation: analytical solutions and numerical simulations[J]. AIMS Mathematics, 2025, 10(11): 25907-25938. doi: 10.3934/math.20251145
In this study, we examined the impact of stochastic factors on the dynamics of nonlinear wave propagation by developing a damped modified Korteweg-de Vries (mKdV) equation incorporating multiplicative noise, mathematically characterized by a Wiener process. Standard deterministic models, although proficient in idealized contexts, frequently fail to accurately depict the complex behaviors elicited by the stochastic fluctuations intrinsic to physical systems. To mitigate this limitation, we utilized a hybrid analytical-numerical approach, employing the modified simple equation method to obtain a spectrum of precise soliton and solitary wave solutions. We ran complementary numerical simulations to find out how different levels of noise affect the time evolution and structural stability of these waveforms. The results showed that solitons keep their structure intact when there is not much noise. Nevertheless, as the noise level grows, the amplitude modulation and potential destabilization become more noticeable. Graphs like density maps and three-dimensional surface plots can be used to see how random changes make waves less predictable. These results demonstrated how crucial it is for nonlinear wave models to include random parts. Future research on complex noise profiles, multi-variable systems, and real-world validation approaches will benefit from this. A stochastic damped mKdV framework, which incorporates multiplicative noise with traditional deterministic or additive-noise analysis, is presented in this article. This paints a fuller picture of the way in which randomness evolves in response to actual system states.
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Mohra Zayed, Hanadi M. AbdelSalam, Ibtisam Daqqa, S. Abdel-Khalek, E A-B Abdel-Salam, Gamal M. Ismail. Stochastic dynamics of solitary waves in the damped mKdV equation: analytical solutions and numerical simulations[J]. AIMS Mathematics, 2025, 10(11): 25907-25938. doi: 10.3934/math.20251145
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