Study On The Exact Solutions, Approximate Solutions And Bifurcation Theory Of Nonlinear Wave Systems

With the development of nonlinear science, more and more scientists believe that our world is nonlinear in essence. The theory of nonlinear system has been involved in almost all of the natural scientific fields. Especially in researching of the modern mathematical and physical problems or scientific engineering problems, many critical problems can be returned to the solutions’ seeking of some special nonlinear systems. Such as the superconductivity, the laser target shooting, the fibre optic communication, the plasma oscillation and the structural phase transition etc. Thus, it is very important to seek the solutions of nonlinear system, especially the solitary wave solutions inclu-ding exact solutions and approximate solutions. These solutions are of great value in scientific researches and applications to clarify the nonlinear wave propagation patte-rn, the algebraic structure and the physical attributes of materials. These solutions are also important for us to explain and forecast many kinds of natural phenomena preci-sely.In this dissertation, we focus on studying the exact solutions, approximate solu-tions and bifurcation theory for nonlinear wave system. Firstly, In chapter 3, we improved several types of major traditional functions expansion method for searching the exact solutions of nonlinear wave systems, which simplified, uniformed and expanded many traditional methods such as the general Riccati equation method,the general G’/G method,the deformation mapping method,the direct algebraic method and the first integral method etc. After using them to three typical types of nonlinear wave system, the VGKdV-mKdV equation with nonlinear terms of any order, the generalized variable-coefficient Gardner equation with forcing term and the coupled Schrodinger-Boussinesq equations,we obtained many significant new solutions except for some known results, including the Jacobi-like elliptical function solutions, solitary-like solutions, triangular-like function solutions and Weierstrass elliptical function solutions etc. The related methods and conclusion has been indexed by the authors all over the worlds. Secondly, In chapter 4, we introduce some common approximate analytical mathods for handing the nonlinear wave system in detail, including the perturbed method, the Adomian decomposition method(ADM), the homotopy analysis method(HAM) and the homotopy perturbation method(HPM), we study the approximate solutions of two important nonlinear wave system including the generalized perturbed KDV-Burgers system and the perturbed Schrodinger equation with variable coefficients by using the HAM and Fourier transform. We analysised the accuracy of some approximate results. Thirdly, In chapter 5, we modify the factional vriational ieration mthod(MFVIM) and use it to solve two kinds of fractional Schrodinger equations with variable coefficients, many kinds of structures for the approximate solutions and exact solutions of them are obtained. And we compared the obtained results with the results obtained by some traditional methods such as HAM, HPM and ADM finally. The analytical results show that the MFVIM enhanced the solving efficiency for the nonlinear system with complex function. Lastly, In chapter 6, by using the bifurcation theory of dynamic system, we analysis the bifurcation structure of a class of generalized fractional Fornberg-Whitham-Rod equation in detail. The whole phase diagram structure under all of the parameters are obtained, and we can deduce some exact and approximate solutions’structure by using these orbits including solitary wave solutions, peakons and periodic wave solutions etc.