Traveling Wave Solutions And Multi-symplectic Structures For Some Nonlinear Partial Differential Equations

In essence,the objective world is nonlinear.Many of so-called linear systems are the results of linear approximation of the nonlinear systems under some specific conditions.The inherent laws and essences of many nonlinear phenomena of the objective world are only described and revealed by the nonlinear models.It is used to describe the nonlinear systems by means of nonlinear partial differential equations.If the exact solutions or numerical solutions of these nonlinear partial differential equations can be obtained,it will be helpful to understand the laws of motion and the essential characteristics of nonlinear systems,to make reasonable explanations for these nonlinear phenomena and to promote the development of mathematics,physics,mechanics and engineering technology.On the one hand,from the historical and current situation of searching for exact solutions of nonlinear partial differential equations,there are many powerful methods,such as the inverse scattering method,B?cklund transformation method,Lie group method,the first integral method,Hirota bilinear method,the homogeneous balance method,the(G?/G)-expansion method,the subsidiary ordinary differential equation method,Jacobi elliptic function method,etc.However,so far,there is not a general method to obtain the exact solutions for any nonlinear partial differential equations.It is still an important task for mathematicians and physicists to explore new methods for seeking the exact solutions of nonlinear partial differential equations.On the other hand,for many nonlinear partial differential equations,it is very difficult to obtain exact solutions or there is no exact solution.In this case,it is natural to resort to numerical methods.In general,the classical difference method,finite element method and spectral method can reach the requirements of the accuracy in a short time.However,these classical methods are difficult to preserve the intrinsic and geometric properties of the nonlinear systems,and it is also difficult to maintain the long-time numerical stability.It is of great importance to study the algorithms which preserve the long-time numerical stability and the intrinsic and geometric properties of these nonlinear systems.In general,people think there is no evident relation between the methods of searching for exact solutions and the methods of obtaining numerical solutions for nonlinear partial differential equations.In this thesis,the techniques and methods of searching for traveling wave solutions of several types of nonlinear partial differential equations are mainly developed.The multi-symplectic structures of some nonlinear partial differential equations are constructed by means of the results of exact solutions.The main content of the thesis can be described as follows:1.The method based on the generalized Riccati-Bernoulli subsidiary ordinary differential equation is presented to construct traveling wave solutions for nonlinear partial differential equations.Nonlinear partial differential equations are converted into a set of nonlinear algebraic equations by using this method.The traveling wave solutions of nonlinear partial differential equations can be obtained by solving the set of nonlinear algebraic equations.Applying this method to a large number of nonlinear partial differential equations,the traveling wave solutions can be obtained.The method can be used to solve the nonlinear partial differential equations and its modified equations by the uniform way,which makes the method more general.The method can be reduced to the Riccati subsidiary ordinary differential equation method and the Bernoulli subsidiary ordinary differential equation method.2.The method based on the Riccati-Bernoulli-elliptic subsidiary ordinary differential equation is presented to construct traveling wave solutions for nonlinear partial differential equations.Nonlinear partial differential equations are converted into a set of nonlinear differential-algebraic equations by means of this method.By using this method,the traveling wave solutions of nonlinear partial differential equations,such as Weierstrass elliptic function solutions,solitary wave solutions and Jacobi elliptic function solutions,can be obtained.The method can be reduced to the(G?/G)-expansion method and Jacobi elliptic function method.3.The method,named the homogeneous balance of undetermined coefficients method,is presented to convert nonlinear partial differential equations into linear equations,bilinear equations and homogeneous equations,respectively.Accordingly,the combined traveling wave solutions,N-soliton solutions and exact traveling wave solutions(hyperbolic function solutions,triangular function solutions and rational function solutions)can be obtained.The(2+1)-dimensional generalized Nizhnik-NovikovVeselov equations are converted into the differently combined bilinear equations by means of the homogeneous balance of undetermined coefficients method.Accordingly,the two types of N-soliton solutions and the explicit exact solutions which are influenced by the specific trivial solutions are obtained by using Hirota bilinear method and the three-wave method,respectively.4.The homogeneous balance of undetermined coefficients method are applied to construct the multi-symplectic structures of some nonlinear partial differential equations,such as Kd V equation,Boussinesq equation,Benjamin-Bona-Mahoney-Camassa-Holm equation,nonlinear sixth-order generalized Boussinesq like equation,BoussinesqBurgers equations and new Hamiltonian amplitude equation,and to construct the generalized multi-symplectic structures of Kd V-Burgers equation and KuramotoSivashinsky equation.The definitions and multi-symplectic structures of Kd V-type equation,Boussinesq-type equation,Boussinesq-Burgers-type equations are given in the multi-symplectic sense.The Kd V equation is chosen to illustrate that the results of T.J.Bridges' method are equivalent to the results of the variational principle,the Euler-Lagrange equation in terms of constructing multi-symplectic structures,law of multi-symplectic conservation,law of local energy conservation and law of local momentum.High precision,long-time numerical stability and preserving the global energy and momentum conservation laws of respective equation,such the advantages of the multi-symplectic algorithm are checked by the numerical results of applying the multi-symplectic Fourier pseudo spectral scheme to Kd V-type equation and BenjaminBona-Mahoney-Camassa-Holm-type equation.