Degree growth of lattice equations defined on a 3x3 stencil

Abstract

We study complexity in terms of degree growth of one-component lattice equations defined on a $3\times 3$ stencil. The equations include two in Hirota bilinear form and the Boussinesq equations of regular, modified and Schwarzian type. Initial values are given on a staircase or on a corner configuration and depend linearly or rationally on a special variable, for example $f_{n,m}={\alpha }_{n,m}z+{\beta }_{n,m}$, in which case we count the degree in $z$ of the iterates. Known integrable cases have linear growth if only one initial values contains $z$, and quadratic growth if all initial values contain $z$. Even a small deformation of an integrable equation changes the degree growth from polynomial to exponential, because the deformation will change factorization properties and thereby prevent cancellations.

arXiv comment19 pages, to appear in "Open Communications in Nonlinear Mathematical Physics", Special Issue in Memory of Decio Levi

References in corpus (15)

• Algebraic entropy
• Singularity confinement and chaos in discrete systems
• Singularity, complexity, and quasi--integrability of rational mappings
• Integrable lattice equations and their growth properties
• Boussinesq-like multi-component lattice equations and multi-dimensional consistency
• Algebraic entropy and the space of initial values for discrete dynamical systems
• Searching for integrable lattice maps using factorization
• A geometric approach to singularity confinement and algebraic entropy
• Investigation into the role of the Laurent property in integrability
• Soliton Taxonomy for a Modification of the Lattice Boussinesq Equation
• On the algebraic structure of rational discrete dynamical systems
• Singularity confinement as an integrability criterion
• Algebraic entropy computations for lattice equations: why initial value problems do matter
• Integrability criterion in terms of coprime property for the discrete Toda equation
• On the singularity structure of the discrete KdV equation

Cited by in corpus (2)

• Algebraic entropy for hex systems
• Algebraic entropy for systems of quad equations