The study of nonlinear stochastic partial differential equations (NSPDEs) constitutes a vital area of research, with extensive applications across diverse fields such as modern physics, biology, superfluid dynamics, image processing, optical fiber communications, plasma physics, and finance. These wide-ranging applications underscore the importance and relevance of NSPDEs in contemporary scientific inquiry.
A central component of stochastic calculus, and one of the most widely studied stochastic processes, is the Wiener process also known as Brownian motion which possesses both martingale and Markov properties. This process is fundamental for modeling random phenomena due to its continuity and normally distributed increments over any time interval. As a result, the Wiener process plays a critical role in the modeling of dispersive systems.
Moreover, there exists a profound and well-established relationship between partial differential equations and stochastic processes, where the stochastic component often introduces randomness into systems governed by deterministic laws. This interplay is particularly evident in the formulation and analysis of NSPDEs, where randomness can model uncertainty or noise in real-world systems. Nonlinear Schrödinger equations are among the most extensively utilized models in applied sciences, owing to their wide spectrum of applications in fields such as optics, fluid dynamics, plasma physics, and quantum mechanics. The exploration of soliton solutions to these equations is of fundamental significance in nonlinear science, as such solutions reveal the physical mechanisms underlying a variety of complex natural phenomena. Over the years, this area has matured into one of the most vibrant and rapidly evolving research domains.
In recent developments, numerous novel solitary wave solutions have been derived using innovative analytical and computational techniques applied to nonlinear models. Particular attention has been given to the study of N-soliton solutions, which serve as a foundation for constructing more intricate wave structures such as lump solutions and rogue waves. These investigations span a wide class of models, including both modified Korteweg-de Vries (mKdV)-type integrable systems and reduced integrable nonlinear Schrödinger-type equations, highlighting the deep interconnectedness of nonlinear wave dynamics across various physical contexts.
The study of nonlinear Schrödinger equations (NLSEs) in the context of optical solitons has emerged as a rapidly expanding area within the field of nonlinear photonics. In recent years, considerable attention has been devoted to examining the effects of various types of nonlinearities on soliton behavior. These include parabolic, Kerr, power-law, polynomial, and saturable nonlinearities, each of which introduces distinct dynamical features into the propagation and stability of optical solitons. These studies are fundamental to the development of advanced optical technologies and to the exploration of wave behavior in intricate nonlinear environments.
Furthermore, the study of stochastic solitons is crucial for understanding the behavior of coherent structures in realistic environments where noise and uncertainty are inherent, such as in optical fibers, fluid dynamics, and plasma physics. Research in this area often involves analytical techniques, numerical simulations, and probabilistic tools to explore how noise affects soliton stability, interaction, and long-time dynamics. In recent years, numerous fields have recognized the effect of including random effects in modeling and analyzing physical processes.
In this work, the stochastic resonant NLSE incorporating both space-time dispersion and intermodal dispersion, with multiplicative noise treated in the Itô sense, together with a generalized Kudryashov-type nonlinearity:
The wave function Q(x, t), which is complex-valued and satisfies , describes the evolution of the wave. The parameters and are considered constant throughout. In Eq. (1), the first term accounts for the linear progression over time, whereas the coefficients and are linked to chromatic dispersion (CD) and space-time dispersion (STD), also referred to as self-steepening effects, respectively. Here, characterizes the intermodal dispersion (IMD), and quantifies the strength of the resonant nonlinearity. The coefficients and (for ) are associated with self-phase modulation (SPM) and related nonlinear effects, while n is the power-law nonlinearity parameter. In this setting, W(t) represents a standard Wiener process and denotes the noise strength. The white noise component is modeled as the formal time derivative .
Generalized Kudryashov's law nonlinearity refers to a mathematical model for the nonlinear refractive index in optical systems, providing a comprehensive framework that allows for multiple terms of arbitrary powers of intensity to be included in the equations describing the propagation of light pulses.
The SRNLSE has been studied by few authors such as, Eslami et al. discussed the optical soliton solutions of the SRNLSE using New Kudryashov's method and . Trouba et al. investigated the optical soliton solutions of SRNLSE by applying the extended auxiliary equation method.
In the current work, the proposed model is studied for the first time using the modified extended mapping technique, through which we derive a rich variety of exact analytical solutions, including dark and singular solitons, as well as singular periodic solutions, hyperbolic solutions, periodic and rational solutions. The influence of stochastic perturbations is further illustrated by comprehensive two- and three-dimensional graphical representations. This dual contribution-diverse explicit solutions combined with visual clarification of stochastic effects-offers deeper theoretical insights and practical guidance on the robustness of soliton transmission in noisy environments.
The paper is organized into five main sections: the next section offers an overview of the proposed approach, followed by the findings in Section "Solitons and other solutions to the proposed model". Section "Visual graphs for some solutions" presents examples of 2D and 3D graphical simulations, including contour plots. The conclusion is found in Section "Conclusions".