Lie Symmetry Analysis And Conservation Law For Hyperbolic Geometric Flow Equations

In this thesis,hyperbolic geometric flow equation and SL’(2)-motion equation of plane curve are studied by applying the classical Lie symmetry method.And the conservation laws of modified hyperbolic geometric flow and dissipative hyperbolic geometric flow are discussed.Firstly,the nonlinear self-adjointness and conservation laws of the modified dissipative hyperbolic geometric flow equation are studied.Secondly,we study the group invariant solutions,global classical solutions and blow up of dissipative hyperbolic geometric flows on Riemann surfaces.Thirdly,we investigate Lie symmetry group,optimal system,exact solutions and conservation laws of modified hyperbolic geometric flow via Lie symmetry method.Finally,by using the theory of group-invariant solutions,the symmetries of the motion equation of plane curve in SL’(2)geometry Group-invariant solutions are presented.One-dimensional optimal system and group-invariant solutions are obtained.In the first part,the nonlinear self-adjointness of the modified dissipative hyperbolic geometric flow equation is studied.Furthermore,based on the method of Ibragimov,we can obtain conservation laws and nonlocal conservation laws for the modified dissipative hyperbolic geometric flow equation.In the second part,we study exact solutions of dissipative hyperbolic geometric flows on Riemann surfaces.In this chapter,the group invariant solutions of dissipative hyperbolic geometrical flow are analyzed by Lie group method.In the process of reduction,elliptic equation and hyperbolic equation are obtained.Then,for the Cauchy problem of the hyperbolic equation,we prove that there always has an initial symmetric tensor such that the global classical solution exists for any given initial metrics.Otherwise,the phenomenon of blow up is discussed.In the third part,we investigate Lie symmetry group,optimal system,and exact solutions of modified hyperbolic geometric flow via Lie symmetry method.Then conservation laws of modified hyperbolic geometric flow are obtained by applying Ibragimov method.In the fourth part,by using the theory of group-invariant solutions,the symmetries of the motion equation of plane curve in SL’(2)geometry are presented.One-dimensional optimal system and Group-invariant solutions are discussed.In the finally part,the content is summarized and prospected.