Atlantis Press Paper Details: A New Discrete Hénon-Heiles System

By considering the Darboux transformation for the third order Lax operator of the Sawada-Kotera hierarchy, we obtain a discrete third order linear equation as well as a discrete analogue of the Gambier 5 equation. As an application of this result, we consider the stationary reduction of the fifth order Sawada-Kotera equation, which (by a result of Fordy) is equivalent to a generalization of the integrable case (i) HénoHeiles system. Applying the Darboux transformation to the stationary flow, we find a Bäcklund transformation (BT) for this finite-dimensional Hamiltonian system, which is equivalent to an exact discretization of the generalized case (i) Hénon-Heiles system. The Lax pair for the system is 3 × 3, and the BT satisfies the spectrality property for the associated trigonal spectral curve. We also give an example of how the BT may be used as a numerical integrator for the original continuous Hénon-Heiles system.

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title:		A New Discrete Hénon-Heiles System
publication:		JNMP
volume-issue:		10 - Supplement 2
pages:		27 - 40
ISSN:		1402-9251
DOI:		doi:10.2991/jnmp.2003.10.s2.3 (how to use a DOI)
author(s):		Alan K COMMON, Andrew N W HONE, Micheline MUSETTE
publication date:		December 2003
abstract:		By considering the Darboux transformation for the third order Lax operator of the Sawada-Kotera hierarchy, we obtain a discrete third order linear equation as well as a discrete analogue of the Gambier 5 equation. As an application of this result, we consider the stationary reduction of the fifth order Sawada-Kotera equation, which (by a result of Fordy) is equivalent to a generalization of the integrable case (i) HénoHeiles system. Applying the Darboux transformation to the stationary flow, we find a Bäcklund transformation (BT) for this finite-dimensional Hamiltonian system, which is equivalent to an exact discretization of the generalized case (i) Hénon-Heiles system. The Lax pair for the system is 3 × 3, and the BT satisfies the spectrality property for the associated trigonal spectral curve. We also give an example of how the BT may be used as a numerical integrator for the original continuous Hénon-Heiles system.
copyright:		© Atlantis Press. This article is distributed under the terms of the Creative Commons Attribution License, which permits non-commercial use, distribution and reproduction in any medium, provided the original work is properly cited.
full text:		10s2_3.pdf (143 K)