Analysis On Solitary Waves And Local Modes Of Discrete Nonlinear Differential-Difference Lattice Systems

Nonlinear Science, which has solition theory, fractal and chaos as its main parts, is the subject of studying the common futures of nonlinearity. Nonlinearity is universal and important. Most nonlinear problems can be described by nonlinear equations, which generally includes nonlinear ordinary differential equations, partial differential equations (PDE), difference equations, functional equations and differential-difference equations (lattice systems). Nonlinear differential-difference lattice system is featured by the discreteness of its part or all spatial variables and the continuity of its time variable.How to obtain the exact solutions of nonlinear equations is of vital importance to the study of the corresponding problem. The key problem in soliton theory is to get solutions of the nonlinear evolution equations, including exact ones or numerical ones. During the past 50 years or so, the scientists have created various ingenious methods to construct exact solutions, especially soliton solutions of nonlinear equations. However, compared to nonlinear partial differential equations, the study of constructing solitary solutions of differential-difference lattice systems started latter and less progress has been made.In this dissertation, along with the direction of the development of soliton theory, centered with discrete soliton lattice systems, under the guidance of mathematics mechanization and by means of computer algebraic system software and the Wu method, some problems of solving some important discrete nonlinear differential-difference lattice systems are discussed and some methods for constructing the exact explicit solutions of differential-difference lattice systems are presented and improved. Many explicit solutions for such lattice systems are obtained. By using the reverse method, we construct discrete models with long-range interactions. Several kind of exact solitary wave solutions are presented for the continuum Toda lattice equation.The history and development of the soliton theory as well as the re- search status of mathematics mechanization are reviewed at the beginning of this dissertation. Then the theory of constructing exact solutions of partial differential equations (PDEs) under the AC = BD theory is introduced. The methods known up to today for constructing exact solutions of nonlinear differential-difference lattice systems are summarized and analyzed. The nonlinear characteristics of discrete integrable Toda lattice are presented. The solitary wave solutions of classical Toda Lattice is introduced. The elastic collision between the double-soliton of Toda lattice is analyzed. Several kinds of solitons in nonlinear lattices with next-neighbor interactions are presented.In the subject of discrete nonlinear differential-difference lattice systems, by introducing negative power terms, and enlarging the scope of the combination function to both hyperbolic functions and triangular functions, we give the modified F—expand, method and the extended tanh-sech method. A general Jacobi ellipse expansion method and the Fibonacci tan-sec-expand method are also presented in this dissertation. By using the above mentioned methods as well as the extended Sine-Gordon method, a lot of discrete equations, such as Toda lattice hierarchy, discrete mKdv lattice systems, Hybrid lattice , Ablowitz-Ladik lattice and Volterra lattice are studied and abundant explicit traveling wave solutions are obtained. By using auxiliary equations, the tanh-expand method is also extend to differential-difference lattice systems with variable coefficients and several exact solutions are given in this case.The study of discrete compacton solutions is the second part of this dissertation. Two discrete Klein-Gordon lattice models with long-range interactions are established by reverse method. The localized modes with first-and-second-neighbor interactions are studied. Discrete breathers as well as more kinds of discrete N-site compactons under the condition of next neighbor interactions are presented. The next-neighbor coefficient has an influence on the stability of N-site compactons. By numerical simulation it is shown that the broad breathers are stable while the narrow ones are not.In the case of nonlinear evolution equations, the continuums Toda lattice model is extended to include coupling of longitudinal and transversal interactions. Supposing that the transversal and the longitudinal strains be of the same order of magnitude , by using direct algebraic method, we obtain compactons, multiple compactons ,peakons and compacton-like solutions of the continuum Toda lattice system . Compactons are zero outside of a finite spatial region, while peakons have cusp at their peak of valley, and the width and the amplitude of compacton-like solutions are determined by the velocity. For the special case of the continuum Toda lattice system, by using the extended sin-cos expand method, more exact traveling wave solutions are given. Profiles of most of those weak excitations and solitary wave solutions are presented for a better understanding.A brief summary is given in the last part of the dissertation, while a possible academic prospect is also opened.