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.

In this study, we investigate the stochastic modified Korteweg-de Vries (SmKdV) equation, which is driven in the Itô sense by advection noise. We show that by solving certain deterministic counterparts of the modified Korteweg-de Vries with an extra diffusion term (for short DmKdV), and then combining the results with a solution of stochastic ordinary differential equations, the exact solution of the SmKdV equation may be discovered. We derive soliton solutions for the DmKdV problem using two distinct methods: the extended tanh function approach and the $\exp (-\psi (\eta ))$ -expansion method. Additionally, we study how the advection noise affects the solutions of the SmKdV equation by presenting several 3D graphs using a MATLAB program.