The Research Of Some Nonlinear Partial Differential Equations' Exact Solutions

In the paper,we study exact solutions for some nonlinear partial differential equations.In the mathematical physics field,many nonlinear differential equations are proposed.However,it is very hard to find the exact solutions to these equations since they are nonlinear equations.In particula r,to construct the exact solutions becomes one of the main problems in nonlinear science.On the one hand,some important methods are provided and developed greatly,such as inverse scattering method,Backlund transformation method and bilinear method and so forth.By these methods,we can not only give single traveling wave solutions,but also multi-traveling wave solutions.On the other hand,a lot of direct methods are introduced to find the exact solutions.Among those,the direct expansion method is used extensively.The main idea is to expand the solution as a special form in terms of some elementary or special functions.Then we substituting it into the equation and give the corresponding coefficients in solutions.By the method,the exact solutions for many nonlinear differential equations are obtained.We use two powerful new methods namely complete discrimination system for polynomial method and trial equation method to find the exact solutions.By the first method,we can give the classification of all single traveling wave solutions for some nonlinear differential equations.If a differential equation can be reduced to an integral form,we can use this method to classify its solutions.But if the equation cannot be reduced to an integral form,w e will use the trial equation method to find a factor equation which is just the so-called trial equation.This trial equation can be reduced to the integral form,and then ca n be solved by complete discrimination system for polynomial method.In the paper,firstly,by the complete discrimination system method,we concretely study the Zhiber-Shabaut equation,Landau-Ginzberg-Higgs equation,Klein-Gordon equation and K(m,n)equation with different parameters,and give complete classifications of their all single traveling wave solutions.Secondly,we use the trial equation method to study the Bretherton equation and deformed Boussioesq equation and give their rich exact traveling wave solutions.