Invariant Subspace Methods And Improved Hyperbolic Tangent And Its Application In Differential Equations

Nonlinearity which caused the infinite diversity, mutability and evolution of the world, is universal in nature. And it is the branches of all of the natural science and social science. As we see, it is important to study nonlinear problem. So people study it with nonlinear equation which is the mathematical model of nonlinear problem. We call nonlinear equation with time-terms nonlinear evolution equation.In order to study nonlinear phenomenon better, we should start it from the exact solutions of these nonlinear equations. So solving method and skill is especially impor-tant. This thesis first introduced several common methods:inverse scattering transform Method (IST), Hirota bilinear method, Backlund transform method, homogeneous bal-ance method and the method of hyperbolic tangent function. On the basis of Malfliet’s work, We introduced improved hyperbolic tangent method for solving nonlinear evolution equation. With this method, we will give the new and more complicated exact solutions of KP equation and KdV equation. In this thesis, we will also introduce the generalized variable separation method-invariant subspace mothod which based on principle of separation of variables and proposed by Sturm and Liouville. In order to verify the effec-tiveness of this method, we will give two examples. One is the general simple nonlinear evolution equation and another is the system of KdV-type equations.For a long time, symbolic computation system plays an important role in getting exact solutions by solving nonlinear partial differential equation. This thesis will simplify the solving process with symbolic computation software Maple. And there is the powerful function of2D and3D plotting in Maple. So we give the2D and3D graphs of the exact solution with this function. It makes the solutions more vivid, and makes us more easily to study the meaning, properties of the solutions.In this thesis, we mainly study from the following several aspects:Section one, we introduced the feasibility and necessity of research.Section two, firstly we discuss the computer algebra system to illustrate that we can solve the complex equations easily with a variety of mathematical software (such as Maple). Next, we will introduce several classic methods of solving nonlinear partial dif-ferential equation,from which we can learn basic idea and principle of these methods and the relationship between them. Then, give the solitary wave theory and see the properties, shape and meaning of solitary wave solutions. At last, proposed some integrabilities from different sense and illustrate the relationship between integrability and the existence of the exact solutions of nonlinear differential equations.Section three Introduced the basic idea and principle of the improved hyperbolic tangent method and its solving steps. finally, solving KP and KdV equations with this method to get their complicated new exact solution. With Maple, plot the2D and3D graph of every solution to verify they are meaningful.Section four, we introduce the principle and solving steps of invariant subspace method. And study the effectiveness of this method by two examples. With Maple, plot the2D and3D graph of every solution to verify the correctness of them.Section five is a summary of this thesis and the prospect of solving nonlinear evolu-tion equations.