Integrable lattice equations and their growth properties

Abstract

In this paper we investigate the integrability of two-dimensional partial difference equations using the newly developed techniques of study of the degree of the iterates. We show that while for generic, nonintegrable equations, the degree grows exponentially fast, for integrable lattice equations the degree growth is polynomial. The growth criterion is used in order to obtain the integrable deautonomisations of the equations examined. In the case of linearisable lattice equations we show that the degree growth is slower than in the case of equations integrable through Inverse Scattering Transform techniques.

References in corpus (1)

• Linearisable Mappings and the Low-Growth Criterion

Cited by in corpus (29)

• Continuous Symmetries of Difference Equations
• Logarithmic difference lemma in several complex variables and partial difference equations
• Algebraic Entropy for lattice equations
• Searching for integrable lattice maps using factorization
• Weak Lax pairs for lattice equations
• On various integrable discretisations of a general two component Volterra system
• Algebraic entropy for differential-delay equations
• Integrable discrete autonomous quad-equations admitting, as generalized symmetries, known five-point differential-difference equations
• Integrable Lattice Maps: $Q_5$, a Rational Version of $Q_4$
• Algebraic entropy computations for lattice equations: why initial value problems do matter
• Algebraic entropy for semi-discrete equations
• Linearizability and fake Lax pair for a consistent around the cube nonlinear non-autonomous quad-graph equation
• Classification of quad-equations on a cuboctahedron
• Algebraic entropy of a class of five-point differential-difference equations
• Algebraic entropy for face-centered quad equations
• Darboux integrable discrete equations possessing an autonomous first-order integral
• Non-Point Invertible Transformations and Integrability of Partial Difference Equations
• Degree growth of lattice equations defined on a 3x3 stencil
• Discrete Painlevé transcendent solutions to the multiplicative type discrete KdV equations
• Algebraic entropy for systems of quad equations
• Algebraic entropy for hex systems
• Reconstructing a Lattice Equation: a Non-Autonomous Approach to the Hietarinta Equation
• A non autonomous generalization of the $Q_{\text{V}}$ equation
• A Lax pair of a lattice equation whose entropy vanishes
• On the three-dimensional consistency of Hirota's discrete Korteweg-de Vries Equation
• Linear degree growth in lattice equations
• Discrete integrable equations on face-centered cubics: consistency and Lax pairs of corner equations
• Properties of the Non-Autonomous Lattice Sine-Gordon Equation: Consistency around a Broken Cube Property
• The Lattice Sine-Gordon Equation as a Superposition Formula for an NLS-Type System