Similarity Reduction To Several Nonlinear Discrete Equations

In this paper，several nonlinear discrete equations are similarity reduced. Themain contents of this paper are as follows.In Chapter1, the emergence and development of soliton theory, the generalmethods of solving the nonlinear equations and the current researching status of thedifferential–difference lattice system are introduced. In Chapter2, the discrete Liepoint symmetry group analysis method is applied on the discrete nonlinearKlein–Gordon equation. Since this equation is not easy to reduce by Lie pointsymmetry method directly, so firstly, this chapter introduces a similaritytransformation to change this equation into a new equation which can be reducedeasily by Lie point symmetry method, then the new equation is reduced by Lie pointsymmetry method and its invariant solutions are gotten. Lastly, by the reversetransformation of the primal similarity transformation, the solutions of the primaldiscrete nonlinear Klein–Gordon equation are gotten. In the second section of Chapter3, by introducing a condition, some exact solutions of Toda lattice equation are gottenunder the given constraints (i.e. the conditional symmetries of Toda lattice equation),these exact solutions are different from those obtained in the past, and thus the scopeof the solutions of this lattice equation is expanded. In the third section of Chapter3, acondition is also introduced (this condition is different from the one in the previoussection), under this condition, we get some new exact solutions of Toda–like latticeequation.