Analytical investigation of a (2+1)-dimensional generalized Schrödinger equation with stochastic perturbations

Abstract

In this research, we conduct an analytical examination of the (2+1)-dimensional generalized nonlinear Schrödinger equation (2D-GNLSE) that includes dispersion, mixed-derivative effects, cubic nonlinearity, and stochastic perturbations. The model we consider enhances the traditional nonlinear Schrödinger framework by incorporating a dispersion coefficient, a nonlinear response coefficient, and a stochastic term. These enhancements facilitate a more accurate representation of nonlinear wave propagation in intricate dispersive and noisy settings, such as optical fibers, plasma waves, and fluid surfaces. Utilizing the proposed unified solver methods, we construct and analyze stochastic solutions. We thoroughly investigate the effects of dispersion and nonlinearity parameters on wave amplitude, localization, and stability, uncovering critical regimes where dispersion and nonlinearity reach a dynamic equilibrium to maintain solitary structures. The stochastic term introduces variations in amplitude and phase, effectively capturing realistic random behavior in nonlinear media. The results obtained illustrate the effectiveness of the unified solver methods in addressing stochastic nonlinear partial differential equations and offer significant insights into the interaction between dispersion, nonlinearity, and randomness in higher-dimensional nonlinear wave systems.

1. Introduction

The nonlinear partial differential equations (NPDEs) play a crucial role in mathematical modeling within contemporary science.^1–3 These equations arise naturally across a variety of disciplines, including superfluid dynamics, nonlinear optics, quantum field theory, and plasma physics, where they govern the evolution of multivariate physical quantities affected by spatial and temporal variations. The inherent nonlinearity of these systems often results in intricate solution structures, such as solitary waves, breathers, shock waves, and blow-up phenomena, which are not present in linear models. A key aim in the exploration of NPDEs is to identify exact or approximate solutions that clarify the physical characteristics of the system being modeled.^4–6 Each method has its own unique set of principles and benefits when applied to a specific problem in order to attain analytical solutions.

The investigation of solitary waves is significantly driven by their extraordinary stability and extensive applications in both science and engineering. These waves maintain their shape and speed over considerable distances, even after colliding with other waves, rendering them effective carriers of energy and information. ⁷ In the realm of fluid dynamics, they account for enduring surface and internal waves; in plasma physics, they represent ion-acoustic and Langmuir structures; in optics, they underpin soliton transmission in fiber communication systems, facilitating high-speed data transfer with minimal loss. In addition to their physical applications, solitary waves also offer profound mathematical insights into nonlinear partial differential equations, emphasizing the relationship between nonlinearity and dispersion. Their capacity to connect theoretical concepts with practical applications renders them a crucial area of research, providing both essential understanding and innovative technological advancements. ⁸

The presence and behavior of solitary waves are fundamentally linked to nonlinear partial differential equations (NPDEs), which provide the essential mathematical framework for their description. Solitary waves generally arise as exact or approximate solutions of nonlinear evolution equations, where a balance exists between dispersion and nonlinearity. Notable models such as the KdV equation, ⁹ the nonlinear Schrödinger equation (NLSE), ¹⁰ coupled cubic NLS-KdV, ¹¹ the Zoomeron equation, ¹² (2+1)-dimensional Pavlov equation, ¹³ and (2+1)-dimensional rdDym equation ¹⁴ exemplify NPDEs that support solitary wave or soliton solutions. These equations encapsulate critical physical processes across various media, such as fluids, plasmas, optical fibers, and quantum systems—where solitary waves manifest as stable, localized entities. The study of NPDEs not only facilitates the prediction and characterization of solitary waves but also enhances understanding of their stability, interactions, and function in energy transport. Therefore, the profound connection between NPDEs and solitary waves underscores the broader importance of nonlinear mathematics in representing real-world nonlinear wave phenomena.

Stochastic partial differential equations (SPDEs) broaden the classical theory of partial differential equations by integrating random effects, thus offering a more realistic framework for modeling systems affected by uncertainty and noise. ¹⁵ These equations emerge naturally in various fields such as fluid dynamics, population biology, financial mathematics, plasma physics, and climate modeling, where randomness is pivotal in the system’s evolution.^16,17 The inclusion of stochastic terms, frequently represented by Brownian motion or Lévy processes, enables SPDEs to account for fluctuations and irregularities that deterministic models are unable to capture. With the rising complexity of real-world systems, SPDEs have emerged as an important study field, combining probability theory and nonlinear analysis while providing compelling models for phenomena where uncertainty is inseparable from dynamics.^18,19

The motivation for studying the Brownian motion process is its central role as the foundation of modern stochastic analysis and its ability to capture uncertainty in a variety of domains. As a quintessential illustration of a continuous-time stochastic process, Brownian motion not only encapsulates the microscopic random movement of particles within fluids but also furnishes a robust mathematical structure for comprehending diffusion, noise, and fluctuations in intricate systems. ¹⁵ Its characteristics render it essential in the formulation of SPDEs that depict phenomena such as heat conduction, plasma dynamics, and random wave propagation. In addition to its applications in physics, Brownian motion serves as the foundation for significant models in finance, including option pricing and risk evaluation, as well as in biology for illustrating molecular transport and population dynamics. By providing a cohesive framework for randomness, Brownian motion inspires research that connects mathematics, natural sciences, and engineering, ultimately promoting both theoretical advancements and practical applications in domains where uncertainty is inherent to system behavior.^20–22

The nonlinear Schrödinger equation (NLSE) is regarded as one of the most significant NPDEs, primarily due to its pivotal role in characterizing wave propagation within dispersive and nonlinear media. ¹⁰ This equation emerges in various fields, including plasma physics, nonlinear optics, fluid dynamics, and Bose–Einstein condensates, where it regulates the behavior of localized structures such as solitary waves and solitons. The NLSE encapsulates the intricate interplay between nonlinearity and dispersion, rendering it an effective model for comprehending stability, collapse, and energy transport in nonlinear systems. From a mathematical standpoint, it acts as a quintessential example for evaluating analytical methods like inverse scattering, variational techniques, and perturbation analysis, in addition to investigating integrability and chaotic dynamics. Its relevance in optical fiber communication systems, where soliton pulses facilitate stable long-distance data transmission, further underscores its practical importance. Consequently, the examination of the NLSE is profoundly driven by both its foundational theoretical depth and its extensive technological and physical applications. The authors investigate the features of oceanic waves using a modified integrable generalized (2+1)-dimensional nonlinear Schrödinger equation system with variable coefficients. ²³ The deterministic (2+1)-dimensional generalized nonlinear Schrödinger equation (2D-GNLSE) with Kerr law ²⁴

$i{\varphi }_t+\alpha {\varphi }_{xx}+{\varphi }_{xy}+{\varphi }_{yy}+\varphi +\gamma \mid \varphi {\mid }^2\varphi =0,i=\sqrt{-1}.$

(1.1)

This form generalizes the 2D-GNLSE with Kerr law by including mixed derivatives ϕ _xy . Here, ϕ(x, y, t) denotes a complex wave profile, α represents the effects of dispersion and γ represents the effects of Kerr nonlinearity. In contrast to the 1D scenario, the 2D-GNLSE allows for more complex dynamics, which encompass self-focusing, beam collapse, modulational instability, and the emergence of localized structures like spatial solitons. These solutions hold considerable significance as they represent the confinement and stability of light beams within nonlinear optical materials. In this work, we analyze model (1.1) in the context of a Brownian motion process, which is outlined as follows:

$i{\varphi }_t+\alpha {\varphi }_{xx}+{\varphi }_{xy}+{\varphi }_{yy}+\varphi +\gamma \mid \varphi {\mid }^2\varphi +\sigma {\beta }_t\varphi =0,$

(1.2)

where σ denotes the strength of noise and β _t characterizes the noise effect, which represents the time derivative of Brownian motion process {β(t)}_t≥0. This stochastic extension is especially important in optical systems, plasma dynamics, and Bose-Einstein condensates, where fluctuations, thermal effects, and external noise all have a large impact on wave stability, soliton production, and energy transmission. The stochastic 2D-GNLSE captures phenomena like noise-induced collapse, stochastic soliton dynamics, and modulation instability under randomness, offering a more realistic framework than the deterministic version. Its study necessitates extensive mathematical techniques from stochastic calculus, numerical simulations, and probabilistic analysis, making it an effective model for comprehending the interaction of nonlinearity, dispersion, and randomness in multidimensional systems.

The unified solver methods (USMs) are the sophisticated and versatile analytical frameworks for finding analytical solutions to the nonlinear partial differential equations,^25,26 including deterministic, stochastic, and fractional forms. The USMs are distinguished by its versatility, as it systematically unifies numerous solution strategies, allowing the creation of a wide range of solution structures. Furthermore, the USMs readily extend to stochastic and fractional systems, allowing researchers to precisely simulate random perturbations and memory effects while visualizing the resultant wave dynamics in multidimensional spaces. The USMs have emerged as a transformative instrument for investigating the rich behavior of nonlinear and complex systems, thanks to its robust, efficient, and generalizable methodology.

In this research, we use the USMs^25,26 to develop a novel substantial stochastic solitary wave solution for the stochastic 2D-GNLSE via the influence of the Brownian motion process in the Stratonovich sense, where chain rule works like classical calculus. We also explain how physical parameters influence the behavior of the solutions. Compared to earlier complicated approaches, USMs have several advantages, including the avoidance of difficult and time-consuming computations and the production of precise results through the utilization of physical factors. They will also be employed as pre-built functions or box solvers to solve various equations and systems encountered in applied science. These solvers are straightforward, reliable, and long-lasting. To our knowledge, these solvers have never been used to address the proposed stochastic model. In fact, these methodologies enable mathematicians, physicists, and engineers to demonstrate some remarkable phenomena in real-world circumstances.

The layout of the current article is described as follows. Section 2 presents reliable analytical methodologies for solving the Duffing equation A Φ″(ζ) + B Φ³(ζ) + C Φ(ζ) = 0. Section 3 provides essential stochastic solutions to the (2+1)-dimensional generalized nonlinear Schrödinger equation. Section 4 outlines the discovered vital solutions. Several graphs illustrating the solutions produced for suitable free parameter values are included. Section 5 presents a conclusion based on proposed results, as well as ideas for further research.

2. The unified solver approaches

The Duffing equation is a key model in the study of nonlinear dynamical systems and oscillatory processes, capturing behaviors that go beyond the purview of linear oscillators. Unlike the classical linear harmonic oscillator, which assumes a proportionate restoring force, the Duffing equation includes nonlinear stiffness effects in its cubic term, allowing for the representation of systems with hardening or softening spring properties. This nonlinearity is required for accurately simulating a wide variety of physical and engineering systems, such as mechanical beams, suspension systems, microelectromechanical devices (MEMS), and electrical circuits. Essentially, the Duffing equation is a prototype model for nonlinear oscillators that offers profound insights into the interaction of external forcing, damping, and nonlinearity. It also serves as a basis for comprehending and designing intricate dynamical systems in applied mathematics, physics, astronomy, and engineering. Here, we examine the following Duffing equation:

$A{\mathrm{\Phi }}^{″}(\zeta )+B{\mathrm{\Phi }}^3(\zeta )+C\mathrm{\Phi }(\zeta )=0.$

(2.1)

The following solutions to equation (2.1) are obtained by using the unified solver approaches outlined in Refs. 25 and 26:

${\mathrm{\Phi }}_{\mathrm{1,2}}(\zeta )=\pm \sqrt{\frac{-2A}{B}}tanh\left(\zeta \right),\frac{A}{B}<0.$

(2.2)

${\mathrm{\Phi }}_{\mathrm{3,4}}(\zeta )=\pm \sqrt{\frac{-2A}{B}}coth\left(\zeta \right),\frac{A}{B}<0,$

(2.3)

C = 2A.

${\mathrm{\Phi }}_{\mathrm{5,6}}(\zeta )=\pm \sqrt{\frac{-2C}{B}}sech\left(\pm \sqrt{\frac{-C}{A}}\zeta \right),\frac{C}{B}<0,\frac{C}{A}<0.$

(2.4)

3. The stochastic solutions

Using the transformation

$\varphi (x,y,t)=\mathrm{\Phi }(\zeta )e^{i(x+y-wt+\sigma \beta (t))},\zeta =x+y-vt,$

(3.1)

v and w represent the speed and frequency of the solitary wave, yields

$\begin{array}{ll}\hfill i{\varphi }_t& =\left(-iv{\mathrm{\Phi }}^{\prime }+\left(\omega -\sigma \frac{d\beta (t)}{dt}\right)\mathrm{\Phi }\right)e^{i(x+y-wt+\sigma \beta (t))},\hfill \\ \hfill {\varphi }_x& =({\mathrm{\Phi }}^{\prime }+i\mathrm{\Phi })e^{i(x+y-wt+\sigma \beta (t))},\hfill \\ \hfill {\varphi }_y& =({\mathrm{\Phi }}^{\prime }+i\mathrm{\Phi })e^{i(x+y-wt+\sigma \beta (t))},\hfill \\ \hfill {\varphi }_{xx}& =({\mathrm{\Phi }}^{″}+2i{\mathrm{\Phi }}^{\prime }-\mathrm{\Phi })e^{i(x+y-wt+\sigma \beta (t))},\hfill \\ \hfill {\varphi }_{xy}& =({\mathrm{\Phi }}^{″}+2i{\mathrm{\Phi }}^{\prime }-\mathrm{\Phi })e^{i(x+y-wt+\sigma \beta (t))},\hfill \\ \hfill {\varphi }_{yy}& =({\mathrm{\Phi }}^{″}+2i{\mathrm{\Phi }}^{\prime }-\mathrm{\Phi })e^{i(x+y-wt+\sigma \beta (t))}.\hfill \end{array}$

Substitute these derivatives into the equation (1.2), yields

$A{\mathrm{\Phi }}^{″}+B{\mathrm{\Phi }}^3+C\mathrm{\Phi }=0,$

(3.2)

from real part, where A = α + 2, B = γ, C = w − α − 1, and v = 2 α + 4 from imaginary part.

Using the suggested unified solvers, the solutions for equation (3.2) are

${\mathrm{\Phi }}_{\mathrm{1,2}}(\zeta )=\pm \sqrt{\frac{-2(\alpha +2)}{\gamma }}tanh\left(\zeta \right),\frac{\alpha +2}{\gamma }<0,$

(3.3)

${\mathrm{\Phi }}_{\mathrm{3,4}}(\zeta )=\pm \sqrt{\frac{-2(\alpha +2)}{\gamma }}coth\left(\zeta \right),\frac{\alpha +2}{\gamma }<0,$

(3.4)

w = 3α + 5.

${\mathrm{\Phi }}_{\mathrm{5,6}}(\zeta )=\pm \sqrt{\frac{-2(w-\alpha -1)}{\gamma }}sech\left(\pm \sqrt{\frac{-(w-\alpha -1)}{\alpha +2}}\zeta \right),\frac{w-\alpha -1}{\gamma }<0,\frac{w-\alpha -1}{\alpha +2}<0.$

(3.5)

Thus, the solutions for equation (1.2) are

${\varphi }_{\mathrm{1,2}}(x,y,t)=\pm \sqrt{\frac{-2(\alpha +2)}{\gamma }}tanh\left(x+y-vt\right)e^{i(x+y-wt+\sigma \beta (t))},\frac{\alpha +2}{\gamma }<0,$

(3.6)

${\varphi }_{\mathrm{3,4}}(x,y,t)=\pm \sqrt{\frac{-2(\alpha +2)}{\gamma }}coth\left(x+y-vt\right)e^{i(x+y-wt+\sigma \beta (t))},\frac{\alpha +2}{\gamma }<0,$

(3.7)

w = 3α + 5.

${\varphi }_{\mathrm{5,6}}(x,y,t)=\pm \sqrt{\frac{-2(w-\alpha -1)}{\gamma }}sech\left(\pm \sqrt{\frac{-(w-\alpha -1)}{\alpha +2}}(x+y-vt)\right)e^{i(x+y-wt+\sigma \beta (t))},$

(3.8)

w − α − 1/γ < 0, w − α − 1/α + 2 < 0.

4. Physical interpretation

The stochastic (2+1)-dimensional generalized nonlinear Schrödinger equation (2D-GNLSE), which includes mixed derivative terms and is influenced by a Brownian motion process, offers a robust framework for simulating randomly perturbed multidimensional nonlinear wave dynamics in realistic physical systems. In practical applications, nonlinear waves found in optical fibers, plasma, or fluid interfaces are seldom deterministic; they develop in settings characterized by noise, fluctuations, and unpredictable external factors. By incorporating a stochastic term driven by Brownian motion into the 2D-GNLSE, one can effectively capture these inherent random effects, including thermal noise, variations in refractive index, or random external potentials, all of which affect the amplitude, phase, and coherence of wave packets. Anisotropic coupling between transverse spatial dimensions is further reflected by the mixed derivative term ϕ _xy , which allows the equation to explain directional dispersion and oblique interactions that take place in complex media. The stochastic 2D-GNLSE provides a comprehensive framework for analyzing the interaction between Brownian fluctuations and spatial anisotropy in shaping the evolution of nonlinear dispersive waves in complex physical systems.

We applied the USMs to generate robust stochastic solutions for the stochastic 2D-GNLSE in the Stratonovich sense. There are several benefits of using the USMs to solve nonlinear partial differential equations. It can effectively generate a variety of solution types, and is incredibly flexible, handling deterministic, stochastic, and fractional systems. The USMs are very strong and dependable techniques in applied mathematics and physics that improve both the comprehension and useful modeling of multidimensional nonlinear systems by offering precise analytical solutions and permitting unambiguous physical interpretation of intricate wave dynamics.

The discovered solutions are essential for comprehending the intricate dynamics of physical systems influenced by random perturbations, noise, or environmental variations In contrast to deterministic solutions, stochastic solutions encapsulate the fundamental uncertainties present in real-world phenomena, including optical pulse propagation in randomly fluctuating media, plasma wave interactions, and the dynamics of Bose–Einstein condensates under thermal or quantum fluctuations. These solutions shed light on phenomena such as noise-induced modulation instability, rogue wave formation, and energy localization in multi-dimensional systems, which classical deterministic methods cannot fully elucidate. Furthermore, stochastic solutions facilitate the creation of probabilistic models for wave behavior, aiding in the prediction, control, and stabilization of nonlinear wave structures in practical scenarios. Ultimately, the integration of stochasticity into the 2D-GNLSE framework reconciles the disparity between idealized theoretical models and the intrinsic randomness of actual physical systems, providing a more thorough and realistic perspective on nonlinear wave dynamics.

To depict the propagation of the solitary waves for the stochastic 2D-GNLSE, 2D charts of chosen solutions are created using Matlab. Figure 1 illustrates the behavior of the developed solution ϕ₁(x, y, t). For σ = 0, the figure present the periodic wave solutions. By increasing σ, the figure depicts stochastic oscillations, with randomness placed on an oscillating pattern. The solution changes over time with inherent unpredictability, possibly caused by Brownian motion or noise.

Figure 1.

Graph of solution ϕ₁(x, y, t) with distinct values of σ for α = 3.5, γ =-0.5.

The dispersion coefficient α in the 2D-GNLSE controls the strength of wave dispersion along the x-direction, affecting how wave packets spread or focus during propagation. Its inclusion is motivated by the need to describe the physical medium’s dispersive features while maintaining a realistic balance of dispersion and nonlinear self-interaction. The parameter α has a direct impact on the shape, stability, and velocity of localized wave structures like solitons. Changing it allows for investigation of alternative propagation regimes in optical fibers, plasmas, and fluid systems. Figures 2 and 3 illustrate the behaviour of periodic and solitary waves, respectively with σ = 0. It was shown that by increasing the parameter α, the amplitude increases with phase shift as in Figure 2. As seen in Figure 3, it was demonstrated that by raising the parameter α, the amplitude increases without phase shift. Similarly, the same behaviour in Figures 4 and 5 in stochastic case, but the solution likely exhibits noise-induced complexity, such as irregular oscillations.

Figure 2.

Graph of solution ϕ₁(x, y, t) with different values of α for γ = -0.5, t = 2 and σ = 0.

Figure 3.

Graph of solution ϕ₅(x, y, t) with different values of α for γ = 1.5, t = 0, w = -4 and σ = 0.

Figure 4.

Graph of solution ϕ₁(x, y, t) with different values of α for γ = -0.5, t = 2 and σ = 2.

Figure 5.

Graph of solution ϕ₅(x, y, t) with different values of α for γ = 1.5, t = 0, w = -4 and σ = 1.

The nonlinearity coefficient γ signifies the intensity of the nonlinear interaction among wave components. This coefficient is driven by the necessity to account for the self-modulation effect, which occurs when the amplitude of the wave affects its own phase or speed of propagation. In a physical context, γ illustrates the nonlinear response of the medium, exemplified by phenomena such as the Kerr effect in optics or self-interaction in plasma and fluid systems. Figures 6 and 7 depicts the behaviour of periodic and solitary waves, respectively with σ = 0. These figures show that by increasing the parameter γ, the amplitude decreases without phase shift. Similarly, the same behaviour in Figures 8 and 9 in stochastic case, but the solution likely exhibits noise-induced complexity, such as irregular oscillations.

Figure 6.

Graph of solution ϕ₁(x, y, t) with different values of γ for α = -3, t = 2 and σ = 0.

Figure 7.

Graph of solution ϕ₅(x, y, t) with different values of γ for α = 3.5, t = 0, w = -4 and σ = 0.

Figure 8.

Graph of solution ϕ₁(x, y, t) with different values of γ for α = -3, t = 2 and σ = 2.

Figure 9.

Graph of solution ϕ₅(x, y, t) with different values of γ for α = 3.5, t = 0, w = -4 and σ = 1.

5. Conclusions

In this paper, we used the unified solver approach to get vital stochastic solutions to the (2+1)-dimensional generalized nonlinear Schrödinger equation governed by Brownian motion. The stochastic formulation accounts for the combined effects of nonlinearity, dispersion, and random disturbances in multidimensional wave systems. As a result, multiple explicit traveling-wave and localized solutions were discovered, illustrating how stochastic forcing changes soliton profiles, modulates their amplitudes, and introduces new wave structures that do not exist in the deterministic scenario. These findings show that random fluctuations have a major impact on wave stability, coherence, and pattern creation in higher-dimensional nonlinear media. The novelty of this study lies in the systematic construction of closed-form stochastic solutions using a unified solver framework and in the clear characterization of noise-induced modifications to multidimensional soliton dynamics. Furthermore, the suggested method is adaptable and uniformly applicable to stochastic and deterministic nonlinear evolution equations, expanding its use to models that emerge in fluid dynamics, plasma physics, and optics.