G. Quispel
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View article: On trilinear and quadrilinear equations associated with the lattice Gel’fand–Dikii hierarchy
On trilinear and quadrilinear equations associated with the lattice Gel’fand–Dikii hierarchy Open
Introduced in Zhang et al. (2012), the trilinear Boussinesq equation is the natural form of the equation for the τ-function of the lattice Boussinesq system. In this paper we study various aspects of this equation: its highly nontrivial de…
View article: On a quadratic Poisson algebra and integrable Lotka-Volterra systems with solutions in terms of Lambert's W function
On a quadratic Poisson algebra and integrable Lotka-Volterra systems with solutions in terms of Lambert's W function Open
We study a class of integrable nonhomogeneous Lotka-Volterra systems whose quadratic terms are defined by an antisymmetric matrix and whose linear terms consist of three blocks. We provide the Poisson algebra of their Darboux polynomials, …
View article: On trilinear and quadrilinear equations associated with the lattice Gel'fand-Dikii hierarchy
On trilinear and quadrilinear equations associated with the lattice Gel'fand-Dikii hierarchy Open
Introduced in 2012, by Zhang, Zhao, and Nijhoff, the trilinear Boussinesq equation is the natural form of the equation for the $τ$-function of the lattice Boussinesq system. In this paper we study various aspects of this equation: its high…
View article: An Elementary Construction of Modified Hamiltonians and Modified Measures of 2D Kahan Maps
An Elementary Construction of Modified Hamiltonians and Modified Measures of 2D Kahan Maps Open
We show how to construct in an elementary way the invariant of the KHK discretisation of a cubic Hamiltonian system in two dimensions. That is, we show that this invariant is expressible as the product of the ratios of affine polynomials d…
View article: Discrete gradients in short-range molecular dynamics simulations
Discrete gradients in short-range molecular dynamics simulations Open
Discrete gradients (DG) or more exactly discrete gradient methods are time integration schemes that are custom-built to preserve first integrals or Lyapunov functions of a given ordinary differential equation (ODE). In conservative molecul…
View article: Measure preservation and integrals for Lotka–Volterra tree-systems and their Kahan discretisation
Measure preservation and integrals for Lotka–Volterra tree-systems and their Kahan discretisation Open
We show that any Lotka–Volterra tree-system associated with an $ n $-vertex tree, as introduced in Quispel et al., J. Phys. A 56 (2023) 315201, preserves a rational measure. We also prove that the Kahan discretisation of these tree-systems…
View article: Trees and superintegrable Lotka-Volterra families
Trees and superintegrable Lotka-Volterra families Open
To any tree on $n$ vertices we associate an $n$-dimensional Lotka-Volterra system with $3n-2$ parameters and, for generic values of the parameters, prove it is superintegrable, i.e. it admits $n-1$ functionally independent integrals. We al…
View article: Measure preservation and integrals for Lotka--Volterra tree-systems and their Kahan discretisation
Measure preservation and integrals for Lotka--Volterra tree-systems and their Kahan discretisation Open
We show that any Lotka--Volterra tree-system associated with an $n$-vertex tree, as introduced in Quispel et al., J. Phys. A 56 (2023) 315201, preserves a rational measure. We also prove that the Kahan discretisation of these tree-systems …
View article: An Elementary Construction of Modified Hamiltonians and Modified Measures of 2D Kahan Maps
An Elementary Construction of Modified Hamiltonians and Modified Measures of 2D Kahan Maps Open
We show how to construct in an elementary way the invariant of the KHK discretisation of a cubic Hamiltonian system in two dimensions. That is, we show that this invariant is expressible as the product of the ratios of affine polynomials d…
View article: Birational maps from polarization and the preservation of measure and integrals
Birational maps from polarization and the preservation of measure and integrals Open
The main result of this paper is the discretization of second-order Hamiltonian systems of the form , where K is a constant symmetric matrix and is a polynomial of degree in any number of variables n . The discre…
View article: Linear Darboux polynomials for Lotka–Volterra systems, trees and superintegrable families
Linear Darboux polynomials for Lotka–Volterra systems, trees and superintegrable families Open
We present a method to construct superintegrable n -component Lotka–Volterra (LV) systems with parameters. We apply the method to LV systems with n components for , and present several n -dimensional superintegrable families. Th…
View article: Linear Darboux polynomials for Lotka-Volterra systems, trees and superintegrable families
Linear Darboux polynomials for Lotka-Volterra systems, trees and superintegrable families Open
We present a method to construct superintegrable $n$-component Lotka-Volterra systems with $3n-2$ parameters. We apply the method to Lotka-Volterra systems with $n$ components for $1 < n < 6$, and present several $n$-dimensional superinteg…
View article: Integrability properties of Kahan's method
Integrability properties of Kahan's method Open
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View article: Discrete gradients in short-range molecular dynamics simulations
Discrete gradients in short-range molecular dynamics simulations Open
Discrete gradients (DG) or more exactly discrete gradient methods are time integration schemes that are custom-built to preserve first integrals or Lyapunov functions of a given ordinary differential equation (ODE). In conservative molecul…
View article: Detecting and determining preserved measures and integrals of birational maps
Detecting and determining preserved measures and integrals of birational maps Open
In this paper we use the method of discrete Darboux polynomials to calculate preserved measures and integrals of rational maps. The approach is based on the use of cofactors and Darboux polynomials and relies on the use of symbolic algebra…
View article: Deducing properties of ODEs from their discretization
Deducing properties of ODEs from their discretization Open
We show that some hard to detect properties of quadratic ODEs (eg certain preserved integrals and measures) can be deduced more or less algorithmically from their Kahan discretization, using Darboux Polynomials (DPs). Somewhat similar resu…
View article: Generalised Manin transformations and QRT maps
Generalised Manin transformations and QRT maps Open
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View article: Homogeneous darboux polynomials and generalising integrable ODE systems
Homogeneous darboux polynomials and generalising integrable ODE systems Open
We show that any system of ODEs can be modified whilst preserving its homogeneous Darboux polynomials. We employ the result to generalise a hierarchy of integrable Lotka-Volterra systems.
View article: A novel 8-parameter integrable map in $\mathbb{R}^4$
A novel 8-parameter integrable map in $\mathbb{R}^4$ Open
We present a novel 8-parameter integrable map in $\mathbb{R}^4$. The map is measure-preserving and possesses two functionally independent 2-integrals, as well as a measure-preserving 2-symmetry.
View article: Homogeneous Darboux polynomials and generalising integrable ODE systems
Homogeneous Darboux polynomials and generalising integrable ODE systems Open
We show that any system of ODEs can be modified whilst preserving its homogeneous Darboux polynomials. We employ the result to generalise a hierarchy of integrable Lotka-Volterra systems.
View article: On the relation between the dual AKP equation and an equation by King and Schief, and its N-soliton solution
On the relation between the dual AKP equation and an equation by King and Schief, and its N-soliton solution Open
The dual of the lattice AKP equation [P.H. van der Kamp et al., J. Phys. A 51, 365202 (2018)] is equivalent to a 14-point equation related to the lattice BKP equation, found by King and Schief. If one of the parameters vanishes, it is equi…
View article: Analogues of Kahan's method for higher order equations of higher degree
Analogues of Kahan's method for higher order equations of higher degree Open
Kahan introduced an explicit method of discretization for systems of first order differential equations with nonlinearities of degree at most two (quadratic vector fields). Kahan's method has attracted much interest due to the fact that it…
View article: Duality for discrete integrable systems II
Duality for discrete integrable systems II Open
We generalise the concept of duality to lattice equations. We derive a novel\n3 dimensional lattice equation, which is dual to the lattice AKP equation.\nReductions of this equation include Rutishauser's quotient-difference (QD)\nalgorithm…
View article: Three classes of quadratic vector fields for which the Kahan discretization is the root of a generalised Manin transformation
Three classes of quadratic vector fields for which the Kahan discretization is the root of a generalised Manin transformation Open
We apply Kahan's discretisation method to three classes of 2-dimensional quadratic vector fields with quadratic, resp cubic, resp quartic Hamiltonians. We show that the maps obtained in this way can be geometrically understood as the compo…
View article: Some integrable maps and their Hirota bilinear forms
Some integrable maps and their Hirota bilinear forms Open
We introduce a two-parameter family of birational maps, which reduces to a family previously found by Demskoi, Tran, van der Kamp and Quispel (DTKQ) when one of the parameters is set to zero. The study of the singularity confinement patter…
View article: On a dual to the lattice AKP equation
On a dual to the lattice AKP equation Open
We present a 3D lattice equation which is dual to the lattice AKP equation. Reductions of this equation include Rutishauser's quotient-difference (QD) algorithm, the higher analogue of the discrete time Toda (HADT) equation and its corresp…
View article: Discrete gradient methods for solving variational image regularisation models
Discrete gradient methods for solving variational image regularisation models Open
Discrete gradient methods are well-known methods of geometric numerical integration, which preserve the dissipation of gradient systems. In this paper we show that this property of discrete gradient methods can be interesting in the contex…
View article: QRT maps and related Laurent systems
QRT maps and related Laurent systems Open
In recent work it was shown how recursive factorisation of certain QRT maps leads to Somos-4 and Somos-5 recurrences with periodic coefficients, and to a fifth-order recurrence with the Laurent property. Here we recursively factorise the 1…
View article: Two classes of quadratic vector fields for which the Kahan\n discretization is integrable
Two classes of quadratic vector fields for which the Kahan\n discretization is integrable Open
Applying Kahan's discretization to the reduced Nahm equations, we obtain two\nclasses of integrable mappings.\n