Degree growth of lattice equations defined on a 3 × 3 stencil

Abstract

<p>We study complexity in terms of degree growth of one-component lattice equations defined on a 3 × 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 = α_n,m z + β_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.</p>