Linear degree growth in lattice equations

Tran, Dinh T.; Roberts, John A. G.

doi:10.3934/jcd.2019023

Cited by 2 publications

(2 citation statements)

References 15 publications

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“…We remark that some alternative degree equations equivalent to (53) are given in [30]. These equivalent forms prove useful for identifying and studying linearisable lattice equations.…”

Section: 2mentioning

confidence: 99%

Algebraic entropy of (integrable) lattice equations and their reductions

Roberts

Tran

2019

Nonlinearity

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We study the growth of degrees in many autonomous and non-autonomous lattice equations defined by quad rules with corner boundary values, some of which are known to be integrable by other characterisations. Subject to an enabling conjecture, we prove polynomial growth for a large class of equations which includes the Adler-Bobenko-Suris equations and Viallet's QV and its non-autonomous generalization. Our technique is to determine the ambient degree growth of the projective version of the lattice equations and to conjecture the growth of their common factors at each lattice vertex, allowing the true degree growth to be found. The resulting degrees satisfy a linear partial difference equation which is universal, i.e. the same for all the integrable lattice equations considered. When we take periodic reductions of these equations, which includes staircase initial conditions, we obtain from this linear partial difference equation an ordinary difference equation for degrees that implies quadratic or linear degree growth. We also study growth of degree of several non-integrable lattice equations. Exponential growth of degrees of these equations, and their mapping reductions, is also proved subject to a conjecture.

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“…We remark that some alternative degree equations equivalent to (53) are given in [30]. These equivalent forms prove useful for identifying and studying linearisable lattice equations.…”

Section: 2mentioning

confidence: 99%

Algebraic entropy of (integrable) lattice equations and their reductions

Roberts

Tran

2019

Nonlinearity

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“…The degree growth of a rational map has been used as a test of integrability, with linear growth associated with particularly simple dynamics. Tran and Roberts [31] establish linear degree growth for several families of mappings, and also find new quad graph mappings with linear growth.…”

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confidence: 99%

Preface Special issue in honor of Reinout Quispel

Celledoni¹,

McLachlan²

2019

Journal of Computational Dynamics

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Reinout Quispel was born on 8 October 1953 in Bilthoven, a small town near Utrecht in the Netherlands. He studied both chemistry and physics, gaining bachelor's degrees at the University of Utrecht in 1973 and 1976 respectively, and then specialized in theoretical physics, with a Master's degree in 1979 (on solitons in the Heisenberg spin chain, supervised by Theodorus Ruijgrok) and a PhD, Linear Integral Equations and Soliton Systems [22], in 1983, supervised by Hans Capel.This thesis, which begins with a study of integrable PDEs, arrives in Chapter 4 (later published in [24]) with the discovery of a method for obtaining fully discrete integrable systems on square lattices, that have as continuum limits the Korteweg-de Vries, nonlinear Schrödinger, and complex sine-Gordon equations, and the Heisenberg spin chain. Thus several of Reinout's lifelong research interests -continuous and discrete integrability, and the relationship between the continuous and the discrete -were present right from the start.The next stop was a postdoc at Twente University, working with Robert Helleman, the founder of the 'Dynamics Days' conference series, before a long-distance move to the Australian National University, working with Rodney Baxter. Reinout and Nel expected this southern sojourn to last for three years; thirty-three years later they are still happily resident in Australia. In 1990 Reinout moved to La Trobe University, Melbourne, where he became a Professor in 2004.Reinout's three main research areas are discrete integrable systems, dynamical systems, and geometric numerical integration, along with interactions between these topics.In discrete integrable systems, having introduced a major new direction in his PhD thesis -his novel reductions to Painlevé equations led to the Clarkson-Kruskal non-classical reduction method -he continued by codiscovering the QRT map [25,26], an 18-parameter family of completely integrable maps of the plane. These turned out to have far-reaching implications in dynamical systems theory, geometry, and integrability. For example, the modern construction of nonautonomous dynamical systems known as discrete Painlevé equations rely on them. Their geometry is explored at length in the 2010 book QRT and Elliptic Surfaces by Hans Duistermaat and is still being investigated today.In dynamical systems, his work has centred on systems with discrete and/or continuous symmetries. His review [28] marked the emergence of reversible dynamical i ii ELENA CELLEDONI AND ROBERT I. MCLACHLAN

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Linear degree growth in lattice equations

Abstract: We conjecture recurrence relations satisfied by the degrees of some linearizable lattice equations. This helps to prove linear growth of these equations. We then use these recurrences to search for lattice equations that have linear growth and hence are linearizable.

Cited by 2 publications

References 15 publications

Algebraic entropy of (integrable) lattice equations and their reductions

Algebraic entropy of (integrable) lattice equations and their reductions

Preface Special issue in honor of Reinout Quispel

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