| In this thesis,several classes of nonlinear wave equations are studied by using the methods of invariant subspaces and dynamical systems.First,a class of generalized third-order KdV equations and a class of space-time fractional Whitham-Broer-Kaup equations are transformed into corresponding ordinary differential equations respectively.Secondly,the bifurcation and phase diagram of the transformed ordinary differential system are drawn by using the dynamical system method and Maple.Bifurcation and phase diagrams are used to obtain the exact solution of the equation,and the image of the partial exact solution is given.Finally,the partial exact solution and its image of a class of nonlinear water wave equations are obtained by using the invariant subspace method.Chapter 1: By introducing the traveling wave transformation to the generalized third-order equation,using the plane dynamic theory and with the help of Maple,the bifurcations and phase diagrams corresponding to the transformed ordinary differential system are given,and the new exact solutions and partial images of the equations are obtained.Chapter 2: By introducing fractional complex transformation to the space-time fractional Whitham-Broer-Kaup equation,transforming it into an integer order ordinary differential equation,and then using the dynamical system method,the vector field of the ordinary differential system in the parameter space is obtained.All the bifurcations and phase diagrams of the equations in different regions are discussed,the solitary wave solutions and periodic solutions of the space-time fractional Whitham-Broer-Kaup equation are obtained.Chapter 3: Using the invariant subspace method to study a class of third-order nonlinear wave equations,the subspaces of the solutions of ordinary differential equations are combined as basis functions,and then the exact solutions of the studied equations are constructed.The invariant subspace method can reduce the nonlinear evolution equation to a finite-dimensional dynamical system,so the invariant subspace method can also be regarded as a dynamical system method in essence. |
