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Title: Theory of integrable lattices
Author: Cheng, Y.
ISNI:       0000 0004 2730 4506
Awarding Body: University of Manchester
Current Institution: University of Manchester
Date of Award: 1987
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This thesis deals with the theory of integrable lattices in "solitons" throughout. Chapter 1 is a general introduction, which includes an historical survey and a short surrunary of the "solitons" theory and the present work. In Chapter 2, we discuss the equivalence between two kinds of lattice AKNS spectral problems - one includes two potentials, while the other includes four. The two nonlinear lattice systems associated with those two spectral problems, respectively is also proved to be equivalent to each other. In Chapter 3, we derive a class of nonlinear differential-difference equations (NDDEs) and put them into the Hamiltonian systems. Their complete integrability are proved in terms of so called "r-matrix". In the end of this Chapter, we study the symmetry properties and the related topics for lattice systems. In particular, we give detail for the Toda lattice systems. Chapter 4 is concerned with the Backlund transformations (BTs) and nonlinear superposition formulae (NSFs) for a class of NDDEs. A new method is presented to derive the generalized BTs and to prove that these BTs are precisely and really the auto-BTs. The three kinds of NSFs are derived by analysis of so called "elementary BTs". In Chapter 5, we investigate some relations between our lattices and the well-studied continuous systems. The continuum limits of our lattice systems and the discretizations of the continuous systems are discussed. The other study is about how we can consider a BT of continuous systems as a NDDE and then how a BT of such a NDDE can be reduced to the three kinds of NSFs of the continuous systems. The last Chapter is a study of integrable lattices under periodic boundary conditions. It provides a mathematical foundation for the study of integrable models in statistical mechanics. We are particularly interested in the lattice sine-Gordon and sinh-Gordon models. We not only prove the integrability of these models but also derive all kinds of classical phase shifts and some other physically interesting relations.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available