Abstract <p>This study derives exact analytical solutions for three forms of the nonlinear Schrödinger equation (NLSE) with multiplicative noise in the Itô sense, using the double-variable expansion method. The selected models include a focusing-type NLSE (NLS<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$^+$</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mrow /> <mrow /> <mo>+</mo> </mmultiscripts> </math></EquationSource> </InlineEquation>), a defocusing-type NLSE (NLS<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$^-$</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mrow /> <mrow /> <mo>-</mo> </mmultiscripts> </math></EquationSource> </InlineEquation>), and a complex cubic NLSE with a delta potential. Through an appropriate wave transformation, the stochastic partial differential equations are reduced to nonlinear ordinary differential equations, enabling systematic application of the expansion method. The framework, constructed via auxiliary functions from a second-order linear ordinary differential equation, facilitates closed-form traveling wave solutions. These include solitonic, periodic, hyperbolic, and rational/rogue-wave-type structures. The method demonstrates robustness in addressing the effects of dispersion, nonlinearity, localized interactions, and stochastic perturbations in a unified analytical framework. Graphical illustrations highlight the dynamic behavior of these solutions under various parameter regimes, including wave number, stochastic noise intensity, and spectral parameters. The derived expressions serve as valuable benchmarks for validating numerical solvers and understanding the modulation and stability of nonlinear wave structures under noise, with potential applications in fiber optics, quantum information, and photonic device modeling. The novelty of this work is the extension of deterministic expansion techniques to stochastic systems, which provides closed-form solutions that accurately capture noise-modulated dynamics. Unlike conventional approaches, such as the inverse scattering transform and Darboux, Bäcklund transformations, the proposed framework accommodates a broader class of nonlinear models and captures richer waveforms. These results reinforce both the theoretical significance and physical relevance of the method, offering practical benchmarks for advancing nonlinear wave research in noisy environments.</p> Graphic abstract <p></p>

Abundant new analytical wave solutions to the Schrödinger equations in optics via the double-variable expansion method

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发表日期：

2025年10月28日

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Md Mamunur Rasid,
Md Nur Hossain,
O. H. Khalil,
K. El-Rashidy,
Wen-Xiu Ma,
Md Mamun Miah

摘要

Abstract

This study derives exact analytical solutions for three forms of the nonlinear Schrödinger equation (NLSE) with multiplicative noise in the Itô sense, using the double-variable expansion method. The selected models include a focusing-type NLSE (NLS $^+$ $^{+}$ ), a defocusing-type NLSE (NLS $^-$ $^{-}$ ), and a complex cubic NLSE with a delta potential. Through an appropriate wave transformation, the stochastic partial differential equations are reduced to nonlinear ordinary differential equations, enabling systematic application of the expansion method. The framework, constructed via auxiliary functions from a second-order linear ordinary differential equation, facilitates closed-form traveling wave solutions. These include solitonic, periodic, hyperbolic, and rational/rogue-wave-type structures. The method demonstrates robustness in addressing the effects of dispersion, nonlinearity, localized interactions, and stochastic perturbations in a unified analytical framework. Graphical illustrations highlight the dynamic behavior of these solutions under various parameter regimes, including wave number, stochastic noise intensity, and spectral parameters. The derived expressions serve as valuable benchmarks for validating numerical solvers and understanding the modulation and stability of nonlinear wave structures under noise, with potential applications in fiber optics, quantum information, and photonic device modeling. The novelty of this work is the extension of deterministic expansion techniques to stochastic systems, which provides closed-form solutions that accurately capture noise-modulated dynamics. Unlike conventional approaches, such as the inverse scattering transform and Darboux, Bäcklund transformations, the proposed framework accommodates a broader class of nonlinear models and captures richer waveforms. These results reinforce both the theoretical significance and physical relevance of the method, offering practical benchmarks for advancing nonlinear wave research in noisy environments.

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