Degree growth of lattice equations defined on a 3 × 3 stencil
: Hietarinta, Jarmo
Publisher: Episciences
: 2024
: Open Communications in Nonlinear Mathematical Physics
: Open Communications in Nonlinear Mathematical Physics
: 2024
: Special Issue 1
: 1
: 19
: 2802-9356
DOI: https://doi.org/10.46298/ocnmp.11589
: https://doi.org/10.46298/ocnmp.11589
: https://research.utu.fi/converis/portal/detail/Publication/484743359
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 fn,m = αn,m z + βn,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.