The Study On Solving Several Classes Of Discrete Systems

In recent years, discrete solitons in several fields of physics including the biological system, atomic chains, solid state physics and photonic structures are investigated. The research for seeking the exact solutions of discrete nonlinear systems has been drawn considerable attention in soliton theory. Therefore it is more important to seek the solutions of the discrete nonlinear system. Besides, the methods of constructing complexiton solutions by using Wronskian technique of nonlinear evolution equations are paid more attention and some further developments are made.The paper includes two main parts. First, it is studied on how to construct the exact solutions of nonlinear difference-differential equations. Second, it is considered how to seek the complexiton solutions with the Wronskian determinants form of nonlinear evolution equations.In chapter 1, new hyperbolic function solutions of the discrete mKdV lattice equation and (2+1)-dimensional Hybrid lattice equation are obtained by introducing new expansion formula of hyperbolic functions. In chapter 2, the method in Ref.[18] is applied to nonlinear difference-differential equations, the discrete mKdV lattice equation and the (2+1)-dimensional Hybrid lattice equation are taken as illustrative examples to find their Jacobi elliptic function solutions. In chapter 3,the generalized Riccati equations method is used to construct many exact solutions including the hyperbolic function solutions and trigonometric function solutions of the discrete (2+1)-dimensional Toda lattice equation and the discrete mKdV lattice equation, and the conclusion contains all results in Ref.[47]. In chapter 4,the new expansion method of three Riccati equations given in Ref.[22] is applied to construct the exact solutions of the nonlinear difference-differential equations, and different types of hyperbolic function solutions of the discrete KdV equation and the discrete mKdV lattice equation are obtained. In chapter 5, the Wronskian determinants method for constructing the complexiton solutions to the Korteweg-de Vries equation is generalized and applied it to obtain the complexiton solutions to the AKNS equation and the Hirota-Statsuma equation. This method can also be applied to construct the exact solutions to other nonlinear evolution equations of which the time part of the Lax pair contains the eigenvalueλ. Hence the method can be used to more equations.