analyzing bifurcation, stability and soliton solutions of wang equation with a multiplicative white noise using hamiltonian and jacobian techniques

The nonlinear schrödinger equation  appears in many fields like quantum mechanics, optical fiber communications, plasma physics, and superfluid dynamics. in this context, we focused on the extended - dimensional stochastic nlse. specifically, we will explore these equations under the influence of multiplicative noise in the itô framework. we apply the sardar sub-equation method to investigate the exact solutions of the extended (3+1) - dimensional stochastic nonlinear schrodinger equation under the influence of multiplicative noise. this method simplifies this nonlinear equation and derive the soliton-like, periodic, bright, dark and singular solutions, which are crucial for understanding wave propagation and stability in various physical systems. in this framework, bifurcation analysis allows us to explore how the system transitions at critical points or parameter thresholds. chaotic behaviors are further examined by adding the external periodic functions. we can characterize regions where chaotic motion emerges, offering insights into unpredictable and turbulent behaviors that are common in plasma physics and optical fibers. sensitivity analysis helps quantify how variations in system parameters influence the dynamics of the equation. by linearizing the system near equilibrium solutions, the stability of critical points is also investigated. moreover, we present the behavior of these solutions graphically. by plotting the solutions obtained from the sardar sub-equation method, we can observe the formation of solitons.  graphical illustrations of bifurcations, chaotic regimes and stability regions to enhance both qualitative and quantitative analysis of the system.