Exact Solutions To Two Nonlinear Partial Differential Equations With Physical Background

Searching for the exact solutions of the nonlinear partial differential equations based onphysical background is helpful to clarify the substance movement and to make a scientificexplanation for the corresponding physical phenomenon。 In the paper, we consider twononlinear partial differential euqaitons arising form fluid mechanics, that is, the modifiedform of Degasperis-Procesi equation which describes shallow water wave models and thegeneralized Vakhnenko equation which describes the propagation of waves in a relaxingmedium. By reducing them into elementary integral forms and by the complete discriminationsystem for polynomial method proposed by Prof. Chengshi liu, we obtain the classificetionsof all single travelling wave solutions of these two equations.