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

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.