Using the bifurcation theory of dynamical systems to the KP-MEW  equation, the existence of solitary wave solutions, periodic cusp wave solutions and other explicit traveling wave solutions are obtained. Under specific parametric condition, various sufficient conditions to guarantee the existence of the above solutions are given. Some exact explicit parametric representations of the above waves are determined.

KP-MEW  equation, solitary wave solutions, periodic cusp wave solutions, loop soliton.