Soliton dynamics in the stochastic nonlinear Schrödinger equation with self-phase modulation and multiplicative white noise.

In this study, we investigate the stochastic nonlinear Schrödinger equation incorporating self-phase modulation under the influence of multiplicative white noise in the dispersionless regime. By employing the improved modified extended tanh function method , we derive a rich spectrum of analytical solutions, including bright and dark solitons, singular and periodic structures, as well as solutions represented through Jacobi and Weierstrass elliptic functions. This analytical framework not only provides a systematic approach for capturing accurate solutions in noisy environments but also provides an effective analytical approach in addressing nonlinear stochastic partial differential equations. We present a thorough graphical analysis that shows solution behavior across various noise intensity regimes and methodically examine the effects of stochastic perturbations on soliton propagation dynamics. The proposed approach provides an analytical framework for constructing exact wave solutions to the considered model and demonstrates its applicability through several representative solution structures.