On the new wave solutions to the MCH equation

We will apply two mathematical methods, namely the exp$(-\phi (\xi \left)\right)$-expansion and sine–cosine in deterministic and stochastic ways for solving the deterministic and stochastic cases of the simplified MCH equation. This nonlinear equation can be turned into another nonlinear ordinary differential equation by appropriate transformation. The methods that we use are efficient and powerful in solving wide classes of nonlinear equations. The solutions obtained are in the form of trigonometric, hyperbolic, exponential and rational functions and a variety of special solutions like kink-shaped, bell-type soliton solutions. Moreover, these solutions reflect some interesting physical interpretation for nonlinear problems. We will also study the stochastic case from the point random wave transformation or some disturbances in the equation itself. The convergence conditions will be discussed.