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Organizers |
Quasi-Hamiltonian Structure, Hojman Construction and Integrable Systems
by
Partha Guha
S.N. Bose National Centre for Basic Sciences
Coauthors: Jose Cariñena and Manuel Rañada
Given a smooth vector field G and assuming the knowledge of an infinitesimal symmetry X, Hojman [J. Phys. A 29 (1996), no. 3, 667-674] proposed a method for finding both a Poisson tensor and a function H such that G is the corresponding Hamiltonian system. We show this construction leads to the degenerate quasi-Hamiltonian structures introduced by Crampin and Sarlet [J.Math.Phys 43 (2002) 2505-2517]. We extend Hojman's construction to Nambu-Poisson case. We give several interesting examples from integrable systems in support of our construction.
Date received: October 28, 2007
Symbolic Computation of Conservation Laws of Nonlinear PDEs in (n+1)-dimensions
by
Willy Hereman
Colarado School of Mines
A direct method will be presented for the symbolic computation of conservation laws of nonlinear PDEs in (n+1)-dimensions. The method computes densities and fluxes based on two key tools: the Euler operator to test exactness and the homotopy operator to invert the total divergence.
The method has been implemented in Mathematica. Using the (2+1)-dimensional shallow-water wave equations as an example, a computer package will be demonstrated that symbolically computes conservation laws of nonlinear PDEs. The software is being used to compute conservation laws of fluid flow (based on the Navier and Kadomtsev-Petviashvili equations) and transonic gas flow.
Date received: September 18, 2007
Leading Order Integrability Conditions for Differential-Difference Equations
by
Mark Hickman
University of Canterbury
A necessary condition for the existence of conserved densities, r, and fluxes of a differential-difference equation which depend on q shifts, for q sufficiently large, is presented. This condition depends on the eigenvalues of the leading terms in the differential-difference equation. It also gives, explicitly, the leading integrability conditions on the density in terms of second derivatives of r.
Date received: September 18, 2007
Integrability and Separation of Variables
by
Ernie Kalnins
University of waikato
Coauthors: W. Miller, K. Kress
A short talk on recent developments in the theory of integrable systems as this relates to the idea of separation of variables is given. An outline of a research programme relating to the idea of superintegrability is briefly discussed with its recent results and future directions outlined.
Date received: October 30, 2007
American Barriers
by
Gerrard Liddell
University of Otago
The problem of managing the finance of multiple projects with early American exercise options can be solved by weak barrier methods. We will describe the symbolic manipulation of the stochastic equations to generate functions for the numerical solution of some of these problems.
Date received: October 31, 2007
Discrete Integrable Systems
by
Reinout Quispel
La Trobe University, Australia
Continuous integrable systems have been known for many years. Examples include the Kepler problem and the Korteweg-de Vries equation. On the contrary, discrete integrable systems, while arguably more fundamental, have only come to the fore in the last 2 decades. We plan to give a brief survey of this new field, concentrating on recent results in the area of discrete integrable mappings (ie integrable nonlinear difference equations).
Date received: August 21, 2007