Integrable Discretizations To A Class Of Soliton Equations

This disquisition mainly consists of five aspects to investigate the in-tegrable discretizations of five soliton equations.The first chapter depicts the historical origin of soliton theory.Chapter 2 introduces some basic concepts, important formulas, essen-tial properties used in this thesis, such as Hirota operator, hyperbolical operator and so forth.Moreover, in chapter 3, we consider the integrable discretization of the famous soliton equations, KdV equation and modified KdV(mKdV) equa-tion. To start with, according to the rational transformation, the above two equations can be transformed into the continuous bilinear equations. After the continuous bilinear equations are substituted by hyperbolic operator, the discrete bilinear equations can be attained. Then we can describe their solion solutions by applying Hirota perturbation method, together with the illumination of integrable property. Lastly, we validate the relations be-tween the discrete KdV equation, mKdV equation and lots of well-known discrete soliton equations, for instance Volterra-Latka(VL) equation, Toda lattice(TL) equation, Self-dual nonlinear network(SDNN) equation. In ad-dition, we apply the Adomian decompose method (ADM) to the SDNN equation to show the significance in numerical analysis. Meanwhile, to the KdV equation, we also consider the discretization in the case of logarithm transformation.Similar to chapter 3, we ponder another two soliton equations in the next chapter. For one thing, we do some research on the focus nonlinear Schrodinger (NLS) equation, and then we reveal the relationship among discrete NLS equation, discrete mKdV equation and the discrete Hirota equation. For the other, the integrable discretization of (Ablowitz-Kaup-Newell-Segur)AKNS equations is also considered. Firstly, we introduce the time development form, spectral problem to the AKNS hierarchy as well as the corresponding Lax pairs and compatible condition. Then by using Hirota method, the soliton solution can be calculated and the inte-grable property can also be proved.Chapter 5 makes the conclusion and expectation to the research, mainly concerning with the feasibility and practicability of the method adopted in the paper.