Item – Theses Canada

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OCLC number
55104996
Link(s) to full text
LAC copy
LAC copy
Author
Tremblay, Sébastien,1973-
Title
Symétries et singularités des équations aux variables discrètes.
Degree
Thèse (Ph. D.)--Université de Montréal, 2002.
Publisher
Ottawa : National Library of Canada = Bibliothèque nationale du Canada, [2003]
Description
2 microfiches
Notes
Comprend des références bibliographiques
Abstract
This thesis is devoted to the study of symmetries and of the integrability of equations with discrete independent variables. In the first three chapters, we introduce a new method which allows us to calculate Lie point symmetries for systems of equations on a lattice. The Lie transformations act simultaneously on the discrete equation and on the lattice itself. The transformations take solutions of the system into other solutions. The case of equations with only one discrete variable is treated first, then we consider the multidimensional case. In Chapter 4, a symmetry classification of possible interactions in a unidimensional diatomic molecular chain is provided. For non-linear interactions, the group of Lie point transformations, leaving the lattice invariant and taking solutions into solutions, is at most 5-dimensional. The equations are classified into equivalence classes under the action of a group of "allowed" transformations. The second part of the thesis is devoted to the question of integrability. In Chapter 5, we investigate this property for partial difference equations with two independent variables. We use newly developed techniques for studying the degree of the iterates. We show that for nonintegrable equations, the degree grows exponentially fast, for integrable lattice equations the degree growth is polynomial. The growth criterion is used in order to obtain integrable deautonomisations of the equations examined. Finally, we show that degree growth contains information that can be an indication as to the precise integration method to be used. In the last Chapter, we examine whether the Painlevé property is a necessary condition for the integrability of nonlinear ordinary differential equations. We show that for a large class of linearisable systems this is not the case. In the discrete domain, we investigate whether the singularity confinement property is satisfied for the discrete analogues of the non-Painlevé continuous systems. Finally, in the Appendix, we find the number and the form of the invariants (generalized Casimir operators) for nilpotent and solvable triangular Lie algebras.
ISBN
0612677877
9780612677876