Study Of Some Problems Via The Symmetry Of Differential Equations

This dissertation focuses on the applications of symmetry methods to study group classification,invariant solutions and conservation laws of some partial differential equations.We can study the properties of partial differential equations by using the symmetry that the equations admitted.We mainly study two classes of partial differential equations.One partial differential equation is Kadomtsev-Petviashvili-Burgers equation,the other is N-th order nonlinear evolution equations.First of all,we study the self-adjointness of the KPBII equation.We show that the equation is nonlinear self-adjoint and it will become strict self-adjoint or weak self-adjoint in some equivalent forms.In addition,by using the Ibragimov's theorem on conservation laws we find some conservation laws for this equation.For the N-th order nonlinear evolution equations,we study Galilei group classification of a general class of nonlinear evolution equations with an arbitrary function by using the classical or special Galilei groups.In addition,we consider invariant solutions and conservation laws of a fourth order variable coefficients Galilei-invariant equation raised from the special Galilei group.We show that the special equation is nonlinear selfadjoint.Moreover,we find that some invariant solutions of the obtained equation can be constructed by the corresponding conservation laws.