Stability, bifurcation, chaotic pattern, phase portrait and exact solutions of a class of semi-linear Schrödinger equations with Kudryashov’s power law self-phase modulation and multiplicative white noise based on Stratonvich’s calculus

Abstract We devote ourselves to finding exact solutions (including perturbed soliton solutions) to a class of semi-linear Schrödinger equations incorporating Kudryashov’s self-phase modulation subject to stochastic perturbations described by multiplicative white noise based on Stratonvich’s calculus. By borrowing ideas of the sub-equation method and utilizing a series of changes of variables, we transform the problem of identifying exact solutions into the task of analyzing the dynamical behaviors of an auxiliary planar Hamiltonian dynamical system. We determine the equilibrium points of the introduced auxiliary Hamiltonian system and analyze their Lyapunov stability. Additionally, we conduct a brief bifurcation analysis and a preliminary chaos analysis of the auxiliary Hamiltonian system, assessing their impact on the Lyapunov stability. Based on the insights gained from investigating the dynamics of the introduced auxiliary Hamiltonian system, we discover ‘all’ of the exact solutions to the stochastic semi-linear Schrödinger equations under consideration. We obtain explicit formulas for exact solutions by examining the phase portrait of the introduced auxiliary Hamiltonian system. The obtained exact solutions include singular and periodic solutions, as well as perturbed bright and dark solitons. For each type of obtained exact solution, we pick one representative to plot its graph, so as to visually display our theoretical results. Compared with other methods for finding exact solutions to deterministic or stochastic partial differential equations, the dynamical system approach has the merit of yielding all possible exact solutions. The stochastic semi-linear Schrödinger equation under consideration can be used to portray the propagation of pulses in an optical fiber, so our study therefore lays the foundation for discovering new solitons optimized for optical communication and contributes to the improvement of optical technologies.