Analyzing Multiplicative Noise Effects on Stochastic Resonant Nonlinear Schrödinger Equation via Two Integration Algorithms

2025-02-14 Fri
stochastic resonance nonlinear Schrödinger equation multiplicative noise solitons wave turbulence
Research Background and Problem IntroductionNonlinear wave systems are core research topics in fields such as physics, optics, and condensed matter physics. However, real-world nonlinear wave systems are often subject to random noise interference, which can significantly alter the behavior of waves, such as soliton propagation, wave turbulence formation, and pattern generation. To more accurately describe these complex phenomena, scientists have proposed the Stochastic Nonlinear Schrödinger Equation (SNLSE) and further developed the Stochastic Resonant Nonlinear Schrödinger Equation (SRNLSE). The SRNLSE incorporates dispersion effects (such as spatio-temporal dispersion and inter-modal dispersion) and nonlinear effects, while introducing stochastic terms to simulate the impact of noise on wave systems.
However, despite the theoretical significance of the SRNLSE, how to solve this equation accurately remains a challenge. Additionally, the impact of noise on the dynamic behavior of nonlinear wave systems still requires in-depth exploration. To address these issues, Khaled A. Gepreel et al. conducted this study, aiming to analyze the effect of multiplicative noise on the SRNLSE through two integration algorithms (Addendum Kudryashov method and Jacobi elliptic expansion method) and explore its application value in soliton dynamics, wave turbulence, and pattern generation.
Paper Source and Author InformationThis paper was co-authored by Khaled A. Gepreel, Reham M. A. Shohib, and Mohamed E. M. Alngar. The authors are from the Department of Mathematics at Taif University in Saudi Arabia, the Basic Science Department at the Higher Institute of Management Sciences and Foreign Trade in Cairo, Egypt, and the Department of Mathematics Education at Sohar University College of Education and Arts in Oman. The paper was published in Volume 57 of Optical and Quantum Electronics in 2025, with article number 156. The DOI is 10.1007/s11082-025-08067-6.
Research Content and Methodsa) Research Process and Experimental Design1. Equation Modeling and Preliminary TransformationThe study starts with the basic form of the SRNLSE, which includes a multiplicative white noise term, Kudryashov’s law, spatio-temporal dispersion (STD), and inter-modal dispersion (IMD). The authors first simplified the equation using the following transformation:
$$ e(x, t) = \Psi(\xi) \exp\left[i(-\kappa x + \omega t + \sigma w(t) - \sigma^2 t)\right], $$
where $\xi = x - vt$, and $\kappa$, $\omega$, and $v$ represent the wave number, frequency, and soliton velocity, respectively. By separating the real and imaginary parts, the authors obtained two key equations (see original formulas (19) and (20)) and determined the IMD coefficient.
2. Addendum Kudryashov MethodThe Addendum Kudryashov method is an analytical technique based on the balance principle for solving nonlinear partial differential equations. In this study, the authors assumed the solution form as:
$$ u(\xi) = \sum_{s=0}^{m} a_s z^s(\xi), $$
where $z(\xi)$ satisfies a specific first-order differential equation. Through homogeneous balancing of $m$, the authors derived the explicit expressions of soliton solutions. For example, when $k = 4\lambda^2$, they obtained bright soliton solutions; when $k = -4\lambda^2$, they obtained singular soliton solutions.
3. Jacobi Elliptic Expansion MethodThe Jacobi elliptic expansion method is a technique that uses Jacobi elliptic functions (such as sn, cn, dn, etc.) to solve nonlinear equations. In this study, the authors assumed the solution form as:
$$ u(\xi) = \sum_{l=0}^{m} \alpha_l [\upsilon(\xi)]^l, $$
and substituted it into the Jacobi elliptic equation:
$$ \upsilon’^2(\xi) = \lambda_4 \upsilon^4(\xi) + \lambda_2 \upsilon^2(\xi) + \lambda_0, $$
to obtain various Jacobi elliptic function solutions (such as sn, cn, dn, and their combined forms). Moreover, when the modulus $m$ approaches 1 or 0, these solutions degenerate into hyperbolic or periodic function solutions.
4. Numerical Verification and Graphical RepresentationTo verify the theoretical results, the authors performed numerical calculations of the equations using Maple software and plotted three-dimensional graphs to illustrate the solution morphology under different parameters. For instance, Figure 1 shows the three-dimensional model of a stochastic bright soliton, while Figure 2 displays the propagation behavior of a stochastic dark soliton.
b) Main Research Results1. Analytical Expressions of Soliton SolutionsThrough the Addendum Kudryashov method, the authors obtained two types of soliton solutions:
- Bright Soliton Solution: When $k = 4\lambda^2$ and $(a - b\beta + \gamma)c_4 > 0$, the solution takes the form:
$$ e(x, t) = \left{-\frac{(n+1)c_3}{2(n+2)c_4} + \sqrt{\frac{4(n+1)(a-b\beta+\gamma)\ln^2 s}{n^2c_4}} \text{sech}(\xi \ln s)\right}^{1/n} e^{i[-\kappa x + \omega t + \sigma w(t) - \sigma^2 t]}. $$
- Singular Soliton Solution: When $k = -4\lambda^2$ and $(a - b\beta + \gamma)c_4 < 0$, the solution takes the form:
$$ e(x, t) = \left{-\frac{(n+1)c_3}{2(n+2)c_4} + \sqrt{-\frac{4(n+1)(a-b\beta+\gamma)\ln^2 s}{n^2c_4}} \text{csch}(\xi \ln s)\right}^{1/n} e^{i[-\kappa x + \omega t + \sigma w(t) - \sigma^2 t]}. $$
2. Jacobi Elliptic Function SolutionsThrough the Jacobi elliptic expansion method, the authors obtained various Jacobi elliptic function solutions, such as:
- When $\upsilon(\xi) = \text{sn}(\xi, m)$ and $(a - b\beta + \gamma)c_4 < 0$, the solution is:
$$ e(x, t) = \left{-\frac{(n+1)c_3}{2(n+2)c_4} + \sqrt{-\frac{(n+1)(a-b\beta+\gamma)m^2}{n^2c_4}} \text{sn}(\xi, m)\right}^{1/n} e^{i[-\kappa x + \omega t + \sigma w(t) - \sigma^2 t]}. $$
- When $m \to 1^-$, the above solution degenerates into a stochastic dark soliton solution.
3. Impact of Noise on Wave SystemsThe study shows that multiplicative noise can significantly alter the propagation behavior of solitons. For example, noise may lead to reduced soliton height, increased occurrence frequency, and randomization of spatial positions. These findings are significant for understanding the role of noise in chaotic systems.
c) Research Conclusions and SignificanceThis study reveals the interaction mechanism between noise and nonlinearity, providing new perspectives for research on soliton dynamics, wave turbulence, and pattern generation. Specifically:
- Scientific Value: The study not only extends the theoretical framework of the SRNLSE but also provides important tools for predicting the behavior of stochastic wave systems.
- Application Value: The research findings have broad application prospects in fields such as nonlinear optics, quantum physics, and condensed matter physics. For example, they can be used to optimize the design of fiber-optic communication systems and improve the stability of optical soliton transmission.
d) Research HighlightsInnovative Methods: For the first time, the Addendum Kudryashov method was combined with the Jacobi elliptic expansion method to solve the SRNLSE.
Diverse Solution Forms: Various solution forms were obtained, including bright solitons, dark solitons, singular solitons, and Jacobi elliptic function solutions.
In-depth Analysis of Noise Effects: The impact of multiplicative noise on soliton propagation behavior was thoroughly explored, revealing the dual role of noise in chaotic systems.
e) Other Valuable InformationThe study also emphasized the potential applications of noise in early warning systems. For example, by combining noise analysis with wave field measurements, the occurrence conditions of extreme waves (rogue waves) can be predicted more accurately. This finding is significant for marine engineering and disaster prevention.
Summary and OutlookKhaled A. Gepreel et al.’s research provides important theoretical support for understanding the impact of noise on nonlinear wave systems. Through innovative mathematical methods and detailed numerical verification, the study not only enriches the solution space of the SRNLSE but also lays a solid foundation for practical applications in related fields. Future research can further explore the role of noise in other complex systems, such as biological networks and fluctuations in financial markets.