A novel unified solver technique for nonlinear partial differential equations with application to the stochastic $ \delta $-nonlinear Schrödinger equation

M. B. Almatrafi; Mahmoud A. E. Abdelrahman; M. B. Almatrafi; Mahmoud A. E. Abdelrahman

doi:10.3934/math.2026243

AIMS Mathematics

2026, Volume 11, Issue 3: 5897-5910. doi: 10.3934/math.2026243

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A novel unified solver technique for nonlinear partial differential equations with application to the stochastic $ \delta $-nonlinear Schrödinger equation

M. B. Almatrafi ¹,
Mahmoud A. E. Abdelrahman ^{1,2
,
,}

1.
Department of Mathematics, College of Science, Taibah University, Madinah, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt

Received: 05 November 2025 Revised: 30 January 2026 Accepted: 06 February 2026 Published: 09 March 2026
MSC : 78A10, 35C07, 60H15, 35Q55, 35R60

A novel and unified solver technique is developed for handling a wide class of nonlinear systems of partial differential equations (NPDEs) that can be systematically reduced to the standard diffusing form with cubic nonlinearity. This canonical structure represents a broad spectrum of nonlinear evolution equations arising in nonlinear optics, superfluids, plasma physics, and quantum field theory. The proposed solver provides a robust analytical framework that efficiently transforms complex NPDEs into solvable ordinary differential forms by applying a proper wave transformation. Its adaptability allows for accurate extraction of solitary, periodic, and stochastic wave solutions under diverse boundary conditions. The solver is primarily used to study the stochastic $ \delta $-nonlinear Schrödinger equation ($ \delta $-NLSE), which incorporates random fluctuations from Brownian motion into nonlinear dispersive dynamics with $ \delta $-type localized perturbations. This application highlights the solver's ability to handle deterministic and stochastic nonlinearities, providing detailed insights into how noise and localized singularities affect the stability and propagation of nonlinear waves in complicated physical mediums. This work presents, for the first time, several analytical solutions to the $ \delta $-NLSE with Brownian noise. The results highlight the accuracy and efficiency of the proposed approach, emphasizing its applicability to address other intricate models in the natural sciences.
- Brownian motion process,
- novel solver technique,
- nonlinear wave propagation,
- $ \delta $-NLSE,
- stochastic analysis
Citation: M. B. Almatrafi, Mahmoud A. E. Abdelrahman. A novel unified solver technique for nonlinear partial differential equations with application to the stochastic $ \delta $-nonlinear Schrödinger equation[J]. AIMS Mathematics, 2026, 11(3): 5897-5910. doi: 10.3934/math.2026243

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Abstract

A novel and unified solver technique is developed for handling a wide class of nonlinear systems of partial differential equations (NPDEs) that can be systematically reduced to the standard diffusing form with cubic nonlinearity. This canonical structure represents a broad spectrum of nonlinear evolution equations arising in nonlinear optics, superfluids, plasma physics, and quantum field theory. The proposed solver provides a robust analytical framework that efficiently transforms complex NPDEs into solvable ordinary differential forms by applying a proper wave transformation. Its adaptability allows for accurate extraction of solitary, periodic, and stochastic wave solutions under diverse boundary conditions. The solver is primarily used to study the stochastic $ \delta $-nonlinear Schrödinger equation ($ \delta $-NLSE), which incorporates random fluctuations from Brownian motion into nonlinear dispersive dynamics with $ \delta $-type localized perturbations. This application highlights the solver's ability to handle deterministic and stochastic nonlinearities, providing detailed insights into how noise and localized singularities affect the stability and propagation of nonlinear waves in complicated physical mediums. This work presents, for the first time, several analytical solutions to the $ \delta $-NLSE with Brownian noise. The results highlight the accuracy and efficiency of the proposed approach, emphasizing its applicability to address other intricate models in the natural sciences.

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