Contact Symmetry Group Analysis For Several Nonlinear Evolution Equations

In recent years,solving nonlinear partial differential equations has become one of the research hotspots because of the rapid development of nonlinear science in various fields.At present,there have been many solutions proposed by some scholars.But,due to the complexity of equations,a unified solution method has not been developed.So we need to explore new solutions or improve the existing methods to make them suitable for solving more complex problems.Based on this purpose,we generalize the concept of optimal system to make it applicable for contact algebras.To illustrate the application of this approach,we study contact symmetries of two second-order nonlinear evolution equations,and establish the one-dimensional optimal systems generated by contact symmetries.In addition,based on the one-dimensional optimal systems,the symmetry reductions for the equations under study are performed.The reduced equations and exact solutions associated with the inequivalent symmetries are obtained.The outline of the dissertation is as follows:Firstly,it briefly describes the research background and current situation of partial differential equations,and introduces some relevant theories about contact symmetry.Secondly,contact symmetry method is used to perform detailed analysis on the first second-order nonlinear evolution equation.In the first step,we have obtained the contact symmetries admitted by the equation,and constructed the one dimension optimal system of the symmetries.In the second step,the reduced equations,invariant solutions and the numerical simulation of invariant solutions corresponding to the optimal system are presented.In addition to the above,by considering the Lie symmetries admitted by the reduced equations,we find out some hidden symmetries of the original equation.Finally,contact symmetry group method is applied to study the second nonlinear evolution equation.Its contact symmetry groups are determined,and optimal system of the symmetries is constructed.In addition,the reduced equations and exact solutions corresponding to the optimal system are presented.