Lie symmetry reductions, bifurcation analysis, and chaotic dynamics of optical bullets in a stochastic nonlinear Schrödinger equation

We investigate a stochastic nonlinear Schrödinger equation modeling optical bullets in dispersive nonlinear media. By applying Lie-symmetry analysis, the governing system is reduced to ordinary differential equations, enabling exact traveling-wave solutions through the Kumar–Malik and Kudryashov approaches. Phase-plane analysis and bifurcation structures are derived, and the Melnikov method is employed to detect homoclinic chaos under stochastic perturbations. The results demonstrate the coexistence of stable solitary structures and chaotic dynamics, offering new insights into energy localization and control in nonlinear optical systems. The methodology unifies symmetry-based reductions, exact solutions, and nonlinear bifurcation analysis, contributing to the broader theory of stochastic nonlinear evolution equations.