Exploration traveling solitary solutions to the modified Korteweg-de Vries equation with advection noise

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.