Invariance properties and symmetry structure of integrable systems

Il Dottorato di Ricerca (PhD) e` stato conseguito in CANADA, University of Waterloo, Waterloo, Ontario, Canada. Tesi depositata alla Biblioteca Nazionale Canadese ISBN: 978-0315-49213-4 L'abstract e` solo in inglese.This thesis is concerned about various aspects of 1 + 1 dimensional non-linear evolution equations. In particular, it is devoted to the study of those properties which can be obtained by the application of non-linear transformations, such as Cole-Hopf, Miura and reciprocal-type transformations. The concept of Backlund Chart is introduced to depict succinctly the various relationships interconnecting different non-linear systems. An account of the Painleve Test, an integrability test for non-linear partial differential equations, is given. The Painleve Analysis indicates how to relate to non-linear integrable systems equations, termed "singularity manifold" equations, the latter exhibit a particular structure which plays a key role in the present investigation. An overview on reciprocal-type transformations follows, especially in connection with their importance in linking non-linear evolution equations. Then, a Backlund Chart which comprises the Caudrey-Dodd-Gibbon (GDG) and Kaup-Kupershmidt (KK) is constructed. It provides an explicit link between such equations and their, respective singularity manifold equations. New hierarchies of integrable non-linear evolution equations are obtained via reciprocal-type transformations. They exhibit a novel invariance and have base member a Kawamoto-type equation. Furthermore, the spatial part of new generic auto-Backlund transformations, for both the CDG and KK hierarchies are constructed. The symmetry structure of a Kawamoto equation is subsequently studied. The Hamiltonian and bi-Hamiltonian formulation of the Kawamoto equation, as well as its hereditary recursion operator are obtained via the links in the Backlund Chart. Subsequently, an analog study is developed for non-linear systems related to the Korteweg-deVries (KdV) equation. An extensive Backlund Chart which incorporates the KdV singularity manifold equation and the KdV "zero soliton" equation is then constructed. It reveals the close analogy between the Harry Dym and Kawamoto equations which turn out to have isomorphic symmetry groups of symmetries. Finally, a new integrability test is proposed which has been termed the Expansion Test. It represents an extension of the Painleve test. The close connection between the singularity manifold equation and the interacting soliton structure follows naturally from the Expansion Test