The Study Of The Group Invariant Solutions And Conservation Laws For Two Kinds Of Nonlinear Wave Equations

The nonlinear wave equation is an important mathematical model of physics and mechanics in natural science and engineering technology.The research on the solving method and the dynamical behavior of solutions has important theoretical and applica-tional significance.In this thesis,we have studied the exact solutions and the conservation laws about a biological chemotaxis model and a class of seven order KdV equation by using the com-bination of Lie symmetry analysis method and the planar dynamical system method,with the help of the Maple software.The specific results are as follows:(1)For a biological chemotaxis model,we obtained its three infinitesimal gen-erators by using the Lie symmetry analysis method.And the corresponding invariant groups and invariant solutions are also obtained.(2)By using the Lie symmetry analysis method,we convert the biological chemo-taxis model into the ordinary differential equations,namely,the wave system.Then we get the explicit expressions and the parameter bifurcation conditions of the kink wave solution,anti-kink wave solutions,hyperbolic traveling wave solutions and periodic wave solutions with the planar dynamical system method.Furthermore,the explicit representations of the three exact solutions with biological significance and the bifur-cation behavior of the original chemotaxis model are obtained.(3)By using the method of conservation laws,which is proposed by Ibragimov,the three conservation laws for the corresponding infinitesimal generators are given.(4)Aiming at a class of seven order KdV equation,we get three infinitesimal gen-erators by using the Lie symmetry analysis method and the Maple,and also get two similarity reduction equations.Then we obtain the exact solutions of the seven order KdV equation.(5)By using the method of conservation laws proposed by Ibragimov,three con-servation laws corresponding to the infinitesimal generators of the seven order KdV equation are obtained.The results of this thesis have a certain theoretical significance.It is helpfull for us to understand the nature of biological chemotaxis by studying a class of bi-ological chemotaxis' exact solutions.The study about the seven order KdV equa-tions' solution and conservation laws helps us to understand the physical phenomena related to traveling wave.