Showing papers by "G. R. W. Quispel published in 2009"

Lagrangian multiform structure for the lattice KP system

Abstract: We present a Lagrangian for the bilinear discrete KP (or Hirota?Miwa) equation. Furthermore, we show that this Lagrangian can be extended to a Lagrangian 3-form when embedded in a higher dimensional lattice, obeying a closure relation. Thus we establish the multiform structure as proposed in Lobb and Nijhoff (2009 J. Phys. A: Math. Theor. 42 454013) in a higher dimensional case.

Closed-form expressions for integrals of traveling wave reductions of integrable lattice equations

Abstract: We give a method to calculate closed-form expressions in terms of multi-sums of products for integrals of ordinary difference equations which are obtained as traveling wave reductions of integrable partial difference equations. Important ingredients are the staircase method, a non-commutative Vieta formula and certain splittings of the Lax matrices. The method is applied to all equations of the Adler–Bobenko–Suris classification, with the exception of Q4.

Linearization-preserving self-adjoint and symplectic integrators

Abstract: This article is concerned with geometric integrators which are linearization-preserving, i.e. numerical integrators which preserve the exact linearization at every fixed point of an arbitrary system of ODEs. For a canonical Hamiltonian system, we propose a new symplectic and self-adjoint B-series method which is also linearization-preserving. In a similar fashion, we show that it is possible to construct a self-adjoint and linearization-preserving B-series method for an arbitrary system of ODEs. Some numerical experiments on Hamiltonian ODEs are presented to test the behaviour of both proposed methods.

Bootstrapping discrete-gradient integral-preserving integrators to fourth order

Abstract: Ordinary differential equations having a first integral may be solved numerically using one of several methods, with the integral preserved to machine accuracy. One such method is the discrete gradient method. It is shown here that the order of the discrete gradient method can be bootstrapped repeatedly to higher orders of accuracy. The potential for improved efficiency offered by the bootstrapped method is illustrated using three 6-dimensional systems.

Structure of B‐series for Some Classes of Geometric Integrators

Abstract: The characterizations of B‐series of symplectic and energy preserving integrators are well‐known The graded Lie algebra of B‐series of modified vector fields include the Hamiltonian and energy‐preserving classes as Lie subalgebras; these spaces are relatively well understood However, two other important classes are the integrators which are conjugate to Hamiltonian and energy‐preserving methods respectively The modified vector fields of such methods do not form linear subspaces and the notion of a grading must be reconsidered We suggest the study of these spaces as filtrations, viewing each element of the filtraton as a trivial vector bundle whose typical fiber replaces the graded homogeneous components In particular, we shall study properties of these fibers; a particular result is that, in the energy‐preserving case, the fiber of degree n is a direct sum of the nth graded component of the Hamiltonian and energy‐preserving space We also give formulas for the dimension of each fiber, thereby providing insight into the range of integrators which are conjugate to symplectic or energy preserving

Lagrangian multiform structure for the lattice KP system

Abstract: We present a Lagrangian for the bilinear discrete KP (or Hirota-Miwa) equation. Furthermore, we show that this Lagrangian can be extended to a Lagrangian 3-form when embedded in a higher dimensional lattice, obeying a closure relation. Thus we establish the multiform structure as proposed in arXiv:0903.4086v1 [nlin.SI] in a higher dimensional case.