Study On Solitary Solutions Of A Class Of Nonlinear Wave Equations

Nonlinearity is universal and important phenomenon in nature. Nonlinear Science, which has soliton, fractal and chaos theories as its main parts, is the subject of studying the nonlinearity. Most nonlinear problems can be described by nonlinear equations. With the recognition of the nonlinear problems, the study on nonlinear systems has become the main topic of domestic and abroad.In the nonlinear systems, the soliton theory of the nonlinear wave equations is the important topic. The study of properties of the solitary waves is of great value in scientific researches and applications to explain the wave propagation and natural phenomena as well as to determine the physical attributes of materials.During the past 50 years, the study on soliton theory of nonlinear wave equation, especially the study on solitary wave solutions, developed quickly and created many methods on the solitary wave solutions of the nonlinear wave equation. Such as, Inverse scattering transformation, Darboux transformation, Backlund transformation, variable separation approach, bilinear method, Truncated Painleve expansion, Clarkson-Kruskal's direct method and so on. In recent years, with the development of computer, the study on solitary wave solution more and more dependents on computer software. Then a series of new methods are obtained and used to study the discrete systems solitary wave solution of nonlinear differential-difference lattice systems. These methods have become the main contend of studying on nonlinear wave equations.In chapter 1 and chapter 2, we introduce the study background, study development and significance of nonlinear wave equation and soliton theory. The methods known up to today for solving the nonlinear wave equation are summarized and analyzed. Then the concerned concepts and theories which used in this paper are introduced.In chapter 3, the singular solitary wave solutions of a kind of nonlinear wave equations are studied. By improved some classical methods, we study solutions of nonlinear wave equation with complex nonlinear terms (double sine-Gordon equation) and find abundant solitary wave solutions(kink solution, anti-kink solution, periodic wave solution) and a kind of new discontinuous solution. Then we prove that the discontinuous solution is discontinuous solitary wave solution by conservation equation theory. The fully nonlinear approximate double sine-Gordon equation is investigated and obtained compacton solution, peakon solution, multi-compacton, multi-peakon solution and discontinuous solitary wave solution. By introducing the concept of nonlinear intensity, the fully nonlinear Klein-Gordon-type equation is researched to find many exact solutions and multi-compacton and singular nonsymmetrical compacton solutions by using the improved generalized Riccati equation methodIn chapter 4, the generalized solutions of generalized KdV equation with variable coefficients are discussed. The many exact solutions, such as, trigonometric function, solitary pattern solutions, solitary wave solutions, Jacobi and Weierstrass elliptic function solutions are given by the auxiliary equation method. A special singular kink solution-nonsymmetrical kink solution is obtained. This method also applies a method to classify the solutions depended various parameters. By compared with other method, this method has the merits of short computation and abundant results, which can be applied to solve other equations with variable coefficients. Then we use the Exp-function method to investigate this equation and obtain generalized solitary wave solution and periodic wave solutions by using mathematic software Mathematica.In chapter 5, the traveling wave solutions of double sine-Gordon equation are investigated by qualitative analysis method. By studying the bifurcation of this equation and dynamic characteristics, we give the expressions of the solitary wave solution and periodic solitary cusp wave solution according with the bifurcation theory. The limit of periodic solitary cusp wave solution and solitary wave solution both equal to the peakon solution. Then the expressions of peakon and anti-peakon solution, periodic solitary wave solution, kink and anti-kink solution are given at various parameters conditions. Some figures are presented by numerical simulation.In chapter 6, the nonlinear differential-difference systems are discussed. By introducing negative power terms and enlarging the scope of the combination function to both hyperbolic functions and triangular functions, which satisfied a Riccati equation. The generalized tanh-sech method is applied to solve the nonlinear differential-difference sine-Gordon equation. Compared with other method, many more solutions of nonlinear differential-difference sine-Gordon equation can be obtained by generalized tanh-sech method. This proves the efficiency of this method in solving the nonlinear differential-difference equations. Then, the F-expansion method is modified to study the nonlinear differential-difference sine-Gordon equation. Many kinds of solitary solutions are obtaioned by using this method.In chapter 7, the summary and expectation are given.