Bifurcation, chaotic and stochastic soliton behavior of the generalized nonlinear Schrödinger equation

The random motion of particles in fluid flow and wave propagation, commonly referred to as Brownian motion, is a well-known physical phenomenon. In this article, soliton solutions of the stochastic generalized nonlinear Schrödinger equation are derived using the modified extended direct algebraic method. To facilitate the analysis, the stochastic partial differential equation is transformed into an ordinary differential equation form. A variety of solutions, including singular-periodic, singular, dark, and dark-bright soliton solutions, are obtained through the proposed method. Graphical representations of these solutions are provided in both two-dimensional and three-dimensional forms to visualize their physical behavior in the presence and absence of stochastic effects. Furthermore, the bifurcation and chaotic dynamics of the generalized nonlinear Schrödinger equation are investigated by constructing the corresponding Hamiltonian and analyzing its phase portrait behavior. The obtained results demonstrate the validity, applicability, and efficiency of the proposed method, and it is expected that these findings will contribute significantly to the study of nonlinear phenomena in diverse fields such as signal processing, control systems, fluid dynamics, communication systems, and financial modeling.