Soliton dynamics in the stochastic nonlinear Schrödinger equation with self-phase modulation and multiplicative white noise

www.nature.com/scientificreports OPEN Soliton dynamics in the stochastic nonlinear Schrödinger equation with self-phase modulation and multiplicative white noise Mohammed F. Shehab1, Hamdy M. Ahmed2 & Hisham H. Hussein3 In this study, we investigate the stochastic nonlinear Schrödinger equation incorporating selfphase modulation under the influence of multiplicative white noise in the dispersionless regime. By employing the improved modified extended tanh function method , we derive a rich spectrum of analytical solutions, including bright and dark solitons, singular and periodic structures, as well as solutions represented through Jacobi and Weierstrass elliptic functions. This analytical framework not only provides a systematic approach for capturing accurate solutions in noisy environments but also provides an effective analytical approach in addressing nonlinear stochastic partial differential equations. We present a thorough graphical analysis that shows solution behavior across various noise intensity regimes and methodically examine the effects of stochastic perturbations on soliton propagation dynamics. The proposed approach provides an analytical framework for constructing exact wave solutions to the considered model and demonstrates its applicability through several representative solution structures. Keywords Soliton solutions, Stochastic nonlinear Schrödinger equation, Wiener process, Improved modified extended tanh-function method. A basic model in mathematical physics, the stochastic nonlinear Schrödinger equation (SNLSE) extends the standard nonlinear Schrödinger equation by incorporating noise effects and random fluctuations1,2. The growing importance of stochastic extensions for explaining wave propagation in physical systems where external noise is significant has motivated considerable research in this area. In this broader context, related stochastic evolution equations, such as the stochastic Burgers equation, have also been investigated using analytical and numerical methods to better understand the role of noise in nonlinear systems3. In modeling wave dynamics in optical fiber communications and other nonlinear media, stochastic effects play a central role in shaping the qualitative behavior of solutions. Building on recent advances in the mathematical theory of stochastic nonlinear Schrödinger equations, these models have proven valuable for applications ranging from optical communication systems to ecological settings under environmental noise4. The three-component nonlinear stochastic Schrödinger equation, for example, has been used to model pulse propagation in birefringent optical fibers subject to Brownian-driven Stratonovich noise, where diverse solution families such as dark, bright, singular, combined, and solitary optical solitons can be obtained, together with descriptions of how their profiles are deformed by the noise5. The extended Klausmeier-type stochastic framework similarly demonstrates how environmental noise reshapes solitary and periodic wave patterns, emphasizing the broader role of stochastic perturbations in modifying soliton structures in nonlinear dispersive systems6. Understanding how noise affects wave propagation remains challenging, because stochastic perturbations can significantly change the qualitative behavior of solutions and generate new emergent features. Recent studies on perturbed and stochastic nonlinear Schrödinger equations show that Wiener-driven noise and higher-order effects can produce diverse soliton families and stability regimes, offering both methodological guidance and physical motivation for studying stochastic nonlinear wave models7. 1Department of Basic Sciences, Faculty of Engineering Technology, ElSewedy University of Technology, Cairo, Egypt. 2Department of Physics and Engineering Mathematics, Higher Institute of Engineering, ElShorouk Academy, El-Shorouk city, Cairo, Egypt. 3School of Mathematical and Computational Sciences, University of Prince Edward Island (UPEI), Cairo Campus, The New Administrative Capital, Egypt. email: Scientific Reports | (2026) 16:16432 | https://doi.org/10.1038/s41598-026-53450-2 1 The phenomenon of self-phase modulation (SPM) in optical systems represents one of the most important nonlinear effects in wave propagation, particularly in the absence of chromatic dispersion8,9. Ultrashort pulse propagation in optical media leads to a rapid modulation of the refractive index through the Kerr nonlinearity. This modulation induces a temporal evolution of the optical phase along the pulse, giving rise to self-phase modulation through the intrinsic coupling between the electromagnetic field and the material response. This phenomenon is important for understanding nonlinear dynamics in photonics applications, particularly in optical fibers and laser systems, because it produces a time-dependent phase shift of optical pulses as they propagate through different media. When chromatic dispersion effects are negligible or absent, SPM becomes the dominant nonlinear mechanism, leading to spectral broadening and temporal pulse evolution that can be modeled through appropriately formulated nonlinear Schrödinger equations8. For this reason, the present model was chosen to study the interaction between stochastic perturbations and self-phase modulation in the dispersionless regime, where the role of SPM can be examined more directly. Multiplicative white noise fundamentally alters nonlinear Schrödinger equation behavior by introducing amplitude-dependent stochastic fluctuations that scale directly with the wave function magnitude, creating interactions that are absent in additive noise scenarios [1,2]. Unlike additive noise, which uniformly affects all components, multiplicative white noise exhibits amplitude-dependent characteristics, where larger amplitude regions experience proportionally stronger perturbations, leading to enhanced instabilities and modified soliton dynamics10. This creates a feedback mechanism in which stochastic perturbations are inherently coupled to deterministic evolution, resulting in phenomena such as noise-induced transitions and changes in qualitative dynamical properties that distinguish these systems from their deterministic counterparts11. Therefore, the inclusion of multiplicative white noise in the present model is physically relevant and mathematically meaningful for describing nonlinear wave propagation in randomly perturbed optical media. Various mathematical techniques have been explored for constructing solutions to nonlinear partial differential equations (NPDEs), ranging from classical analytical methods to modern computational approaches. Among these, the Bilinear Neural Network Method and its improved variant have emerged as efficient methodologies that integrate machine learning to extract complex solutions such as rogue waves and lumps in systems including graphene sheets and shallow water models12,13. 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