Showing papers by "G. R. W. Quispel published in 2012"
TL;DR: This work gives a systematic method for discretizing Hamiltonian partial differential equations (PDEs) with constant symplectic structure, while preserving their energy exactly.
268 citations
TL;DR: In this article, it was shown that Kahan's discretization of quadratic vector fields is equivalent to a Runge-Kutta method, which produces large classes of integrable rational mappings in two and three dimensions.
Abstract: We show that Kahan's discretization of quadratic vector fields is equivalent to a Runge--Kutta method. When the vector field is Hamiltonian on either a symplectic vector space or a Poisson vector space with constant Poisson structure, the map determined by this discretization has a conserved modified Hamiltonian and an invariant measure, a combination previously unknown amongst Runge--Kutta methods applied to nonlinear vector fields. This produces large classes of integrable rational mappings in two and three dimensions, explaining some of the integrable cases that were previously known.
61 citations
TL;DR: In this paper, a method of determining a Lax representation for similarity reductions of autonomous and non-autonomous partial difference equations is presented, which can be used to obtain Lax representations that are general enough to provide the Lax integrability for entire hierarchies of reductions.
Abstract: We present a method of determining a Lax representation for similarity reductions of autonomous and non-autonomous partial difference equations. This method may be used to obtain Lax representations that are general enough to provide the Lax integrability for entire hierarchies of reductions. A main result is, as an example of this framework, how we may obtain the q-Painlev\'e equation whose group of B\"acklund transformations is an affine Weyl group of type E_6^{(1)} as a similarity reduction of the discrete Schwarzian Korteweg-de Vries equation.
21 citations
TL;DR: In this article, the authors derive an nth order difference equation as a dual of a simple periodic equation, and construct explicit integrals and integrating factors of this equation in terms of multi-sums of products.
Abstract: We derive an nth order difference equation as a dual of a very simple periodic equation, and construct ⌊(n + 1)/2⌋ explicit integrals and integrating factors of this equation in terms of multi-sums of products. We also present a generating function for the degrees of its iterates, exhibiting polynomial growth. In conclusion we demonstrate how the equation in question arises as a reduction of a system of lattice equations related to an integrable equation of Levi and Yamilov. These three facts combine to suggest the integrability of the nth order difference equation.
16 citations
Posted Content•
TL;DR: In this paper, a method of determining a Lax representation for similarity reductions of autonomous and non-autonomous partial difference equations is presented, which can be used to obtain Lax representations that are general enough to provide the Lax integrability for entire hierarchies of reductions.
Abstract: We present a method of determining a Lax representation for similarity reductions of autonomous and non-autonomous partial difference equations. This method may be used to obtain Lax representations that are general enough to provide the Lax integrability for entire hierarchies of reductions. A main result is, as an example of this framework, how we may obtain the $q$-Painlev\'e equation whose group of B\"acklund transformations is an affine Weyl group of type $E_6^{(1)}$ as a similarity reduction of the discrete Schwarzian Korteweg-de Vries equation.
1 citations