Degree growth of lattice equations defined on a 3x3 stencil

TL;DR

This paper investigates the degree growth and complexity of lattice equations on a 3x3 stencil, revealing how integrability relates to polynomial growth and how deformations lead to exponential complexity.

Contribution

It characterizes the degree growth behavior of various lattice equations, including integrable and deformed cases, highlighting the impact of initial conditions and deformations on complexity.

Findings

Integrable cases exhibit linear or quadratic degree growth.

Deformations cause a shift from polynomial to exponential growth.

Initial conditions influence the degree growth pattern.

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.