The Investigation Into The Exact Solutions To Some Soliton Equations

Finding exact solutions of the discrete soliton equations is the frontier of science and technology and an important part of the nonlinear science. The discrete soliton equations are used to describe the phenomena and the dynamic processes in many scientific fields, such as particle vibrations in lattices, currents in electrical networks, etc, and these equations play an important role in the fields such as the discretization problem in solid state, queue problem, and numerical simulation of nonlinear partial differential equations. So far, there are still many problems in the high-dimensional and high-order discrete soliton equations.According to the homogeneous balance principle, the G′/G-expansion method is firstly applied to explore the coupled discrete nonlinear Schr?dinger equations and two kinds of the relativistic Toda lattice equations with the aids of the symbolic computation in this thesis, and then the high-order and high-dimensional discrete nonlinear difference-differential equations such as the (2+1)-dimensional Ablowitz-Ladik (AL-NLS) equation and quintic discrete nonlinear Schr?dinger equation are investigated using the G′/G-expansion method. The hyperbolic function solitary wave solutions, the trigonometric function periodic wave solutions and the rational wave solutions with more arbitrary parameters for these equations are derived. When some special values of these parameters are taken, the results can be found in the articles published, thus more abundant exact solutions in this thesis are obtained.These exact solutions have important theoretical significance and scientific value. With the aids of the exact solutions, many phenomena described by the discrete soliton equations can be understood well, such as the exciton motion in molecular crystals, the propagation of discrete self-trapped beams in weakly-coupled nonlinear optical waveguides, the Bose-Einstein condensation in the periodical potential, the energy storage and transmission in the molecular chain in solid physics, nonlinear optics, condensed matter physics, biophysics, etc. In chapter 1, the theoretical significance and scientific value of the nonlinear differential-difference equations are firstly reviewed, and then the development of nonlinear differential-difference equations and main contents of this thesis are outlined.In chapter 2, the main steps of solving nonlinear evolution equation is introduced using the G′/G-expansion method. From the derivation of exact solutions to KdV equation using the G′/G-expansion method, it is easy to see that the G′/G-expansion method is direct,concise,basic,effective, etc.In chapter 3, the coupled discrete nonlinear Schr?dinger equations and two kinds of the relativistic Toda lattice equations are firstly studied by the G′/G-expansion method, and then the (2+1)-dimensional Ablowitz-Ladik (AL-NLS) equation and quintic discrete nonlinear Schr?dinger equation are researched. Three kinds of the exact solutions of nonlinear differential-difference equations above are derived.In chapter 4, the major conclusions of this thesis are given, and the prospects of the exact solution for nonlinear evolution equations are analysed.