Closed-form solutions of stochastic solitary waves for certain type of nonlinear Schrödinger equation

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.