Bifurcation analysis and stochastic optical solutions to the stochastic nonlinear Kodama equation in nonlinear optics

We consider the stochastic nonlinear Kodama equation (SNLKE) driven by multiplicative white noise. A specific wave transformation is applied to convert this system into a one-dimensional conservative Hamiltonian system. We analyze the bifurcation of this system and present its phase portrait. Additionally, a brief description of the phase portrait is provided, along with an illustration of the phase orbit degeneracy depending on the bifurcation parameter. Bifurcation allows us to deduce that changing the parameter values can have a substantial influence on nonlinear optics and mathematical physics as well as the dynamics of the optical soliton solutions of the Kodama equation. Using the conserved quantity, we derive new traveling wave solutions for the SNLKE. In the absence of noise, we recover certain wave solutions for the deterministic case. Furthermore, we examine the influence of multiplicative white noise on the exact solutions of the SNLKE, with some of the obtained solutions visualized graphically.