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Title: Integrable systems and their finite-dimensional reductions
Author: Hone, Andrew N. W.
Awarding Body: University of Edinburgh
Current Institution: University of Edinburgh
Date of Award: 1996
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The first chapter introduces some of the important concepts and structures associated with integrability, and includes a brief overview of some of the applications of integrable systems and their reductions in field theory. Chapter 2 describes the scaling similarity reductions of the Sawada-Kotera, fifth-order KdV, and Kaup-Kupershmidt equations. Similarity solutions of these evolution equations satisfy certain ODEs which are naturally viewed as fourth-order analogues of the Painlevé transcendents; they may also be written as non-autonomous Hamiltonian systems, which are time-dependent generalizations of the integrable Hénon-Heiles systems. The solutions to these systems are encoded into a tau-function, and Bäcklund transformations are presented which allow the construction of rational solutions and some other special solutions. The third chapter is concerned with the motion of the poles of singular solutions (especially rational solutions) of the NLS equation. It is demonstrated that the linear problem for NLS admits an analogue of the well-known Crum transformation for Schrödinger operators, leading to the construction of a sequence of rational solutions. The poles and zeros of these rational solutions are found to satisfy constrained Calogero-Moser equations, and some other singular solutions are also considered. Much use is made of Hirota's bilinear formalism, as well as a trilinear form for NLS related to its reduction from the KP hierarchy. The final chapter deals with soliton solutions of the An(1) affine Toda field theories. By writing the soliton tau-functions as determinants of a particular form, these solutions are related to the hyperbolic spin Ruijsenaars-Schneider system. These results generalize the connection between the ordinary (non-spin) Ruijsenaars-Schneider model and the soliton solutions of the sine-Gordon equation.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available