In this study, we examined a (2+1)-dimensional generalized breaking soliton system (GBSS) using both analytical and numerical methods. By applying a generalized direct algebraic method, we derived exact solutions that displayed a variety of solitary and periodic wave patterns. These solutions illuminated the interplay between nonlinearity and dispersion in several physical contexts, including fluid dynamics, plasma physics, and nonlinear optics. In addition, we developed a robust numerical scheme employing an adaptive moving mesh technique based on the MMPDE5 framework. Stability and error analyses confirmed that this method concentrated grid points around steep gradients, achieved second-order spatial convergence, and enhanced computational efficiency. By comparing numerical and exact solutions, we provided more profound insights into GBSS dynamics and facilitated future investigations of complex, multidimensional, and nonlinear wave phenomena.

This study investigates the stochastic nonlinear Schrödinger equation with a delta potential ( $\delta$-NLSE), a model capturing the combined effects of nonlinear dispersion, localized defects, and environmental randomness. Analytical solutions are constructed using unified solver techniques to evaluate the influence of noise intensity and potential strength on wave propagation and solitary-wave formation. The solitary waves in the stochastic $\delta$-NLSE reveal how randomness and defect-induced localization interact, resulting in novel phenomena such as stochastic modification of transmission and reflection coefficients, noise-stabilized limit states, and random soliton diffusion. The issue is significant because real-world media are rarely homogeneous or noise-free, and understanding how randomness interacts with localized singularities remains a challenge in both theory and practice. The proposed stochastic solutions are ground-breaking and highly relevant for modeling complex physical processes in nonlinear wave theory, quantum mechanics, water waves, nonlinear optics, and Bose-Einstein condensates, where localized impurities or interfaces strongly affect wave propagation.