Showing papers by "G. R. W. Quispel published in 2016"
TL;DR: In this paper, Poisson brackets of certain classes of mappings obtained as general periodic reductions of integrable lattice equations are derived from a Lagrangian, using the so-called Ostrogradsky transformation.
Abstract: 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 from a Lagrangian, using the so-called Ostrogradsky transformation. The (q,−p) reductions are (p + q)-dimensional maps and explicit Poisson brackets for such reductions of the discrete KdV equation, the discrete Lotka–Volterra equation, and the discrete Liouville equation are included. Lax representations of these equations can be used to construct sufficiently many integrals for the reductions. As examples we show that the (3,−2) reductions of the integrable partial difference equations are Liouville integrable in their own right.
16 citations
TL;DR: In this paper, Liouville and superintegrability of a generalized Lotka-Volterra system and its Kahan discretization was shown. But the superintegrality of the system was not proven.
Abstract: We prove the Liouville and superintegrability of a generalized Lotka-Volterra system and its Kahan discretization.
11 citations
TL;DR: In this paper, Liouville and superintegrability of a generalized Lotka-Volterra system and its Kahan discretization was proved for general linear systems.
Abstract: We prove the Liouville and superintegrability of a generalized Lotka–Volterra system and its Kahan discretization.
10 citations
TL;DR: In this paper, the 5-point central difference scheme is applied to the semi-linear wave equation and backward error analysis is used to study the solutions of equation through a simple ODE that describes the behavior to arbitrarily high order.
Abstract: How well do multisymplectic discretisations preserve travelling wave solutions? To answer this question, the 5-point central difference scheme is applied to the semi-linear wave equation. A travelling wave ansatz leads to an ordinary difference equation whose solutions can be compared to travelling wave solutions of the PDE. For a discontinuous nonlinearity the difference equation is solved exactly. For continuous nonlinearities the difference equation is solved using a Fourier series, and resonances that depend on the grid-size are revealed for a smooth nonlinearity. In general, the infinite dimensional functional equation, which must be solved to get the travelling wave solutions, is intractable, but backward error analysis proves to be a powerful tool, as it provides a way to study the solutions of equation through a simple ODE that describes the behavior to arbitrarily high order. A general framework for using backward error analysis to analyze preservation of travelling waves for other equations and ...
9 citations
TL;DR: It is shown that despite this result, symplectic Runge-Kutta methods can be volume preserving for a much larger class of vector fields than Hamiltonian systems, and discussed how some Runge -KutTA methods can preserve a modified measure exactly.
3 citations
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TL;DR: In this article, the authors promote the use of discrete gradient methods in image processing by exhibiting experiments with nonlinear total variation (TV) deblurring, denoising, and inpainting.
Abstract: Discrete gradient methods are well-known methods of Geometric Numerical Integration, which preserve the dissipation of gradient systems. The preservation of the dissipation of a system is an important feature in numerous image processing tasks. We promote the use of discrete gradient methods in image processing by exhibiting experiments with nonlinear total variation (TV) deblurring, denoising, and inpainting.