arXiv subjects
Publications and source records attributed to G. Reinout W. Quispel.
In this paper, we present Poisson brackets of certain classes of mappings obtained as general periodic reductions of integrable lattice equations. The Poisson brackets are derived using the so called Ostrogradsky transformation.
A three-step method due to Nijhoff and Bobenko & Suris to derive a Lax pair for scalar partial difference equations (PΔEs) is reviewed. The method assumes that the PΔEs are defined on a quadrilateral, and consistent around the cube. Next, the method is extended to systems of PΔEs where one has to carefully account for equations defined on edges of the quadrilateral. Lax pairs are presented for scalar integrable PΔEs classified by Adler, Bobenko, and Suris and systems of PΔEs including the integrable 2-component potential Korteweg-de Vries lattice system, as well as nonlinear Schroedinger and Boussinesq-type lattice systems. Previously unknown Lax pairs are presented for PΔEs recently derived by Hietarinta (J. Phys. A: Math. Theor., 44, 2011, Art. No. 165204). The method is algorithmic and is being implemented in Mathematica.
Foliate systems are those which preserve some (possibly singular) foliation of phase space, such as systems with integrals, systems with continuous symmetries, and skew product systems. We study numerical integrators which also preserve the foliation. The case in which the foliation is given by the orbits of an action of a Lie group has a particularly nice structure, which we study in detail, giving conditions under which all foliate vector fields can be written as the sum of a vector field tangent to the orbits and a vector field invariant under the group action. This allows the application of many techniques of geometric integration, including splitting methods and Lie group integrators.