Applications Of Elliptic Functions In Discrete Integrable Systems

The objective of this thesis is to introduce the connections between discrete integrable systems and the theory of elliptic functions.We will mainly study the elliptic soliton solutions for discrete integrable systems and the elliptic integrable systems.This thesis starts with an introductory chapter on discrete integrable sistems and aspects of elliptic functions and elliptic curves and their connections.Chapter 2 recalls some useful formulas and analyzies the zeros and poles of some elliptic functions.These will help us to find the discrete elliptic dispersion relations and to construct elliptic direct linearization method.We also introduce two special matrices:lower triangular Toeplitz matrix and skew triangular Hankel matrix,which are used to construct exact solutions for elliptic integrable systems.In Chapter 3,we take the lattice BSQ system as an example to study the discrete elliptic dispersion relations.We apply auto-Backlund transformations to derive the elliptic one soliton solutions for the lattice BSQ system and then we define the elliptic discrete plane wave factor which shows the discrete elliptic dispersion relations.After obtaining the discrete elliptic dispersion relations,Chapter 4 construct-s elliptic direct linearization method which gives the lattice KP system as well as its elliptic soliton solutions.In order to obtain the lattice potential KdV equation and discrete BSQ systems from the lattice KP system,we introduce a new con-cept,namely the elliptic analogues of roots of unity.The lattice potential KdV equation and BSQ system are related to the elliptic square and cubic roots of unity respectively.Chapter 5 focuses on the elliptic potential KdV lattice system,which is a two-parameter extension of the lattice potential KdV equation.When the elliptic curve degenerates,we obtain the lattice potential KdV system.In this chapter we use the generalized Cauchy matrix approach,in particular,involving a Sylvester equation depending on an elliptic curve.Not only the the elliptic potential KdV lattice system but also the the elliptic potential KdV system is derived.We also obtain the exact solutions and Lax pairs to these systems.Moreover,we consider the continuum limits of the elliptic potential KdV lattice system,and rederive the corresponding continous systems.We analyze the dynamics of the soliton solutions for the continous systems and find some new features of the elliptic system in comparison to the non-elliptic case.