Degree growth of lattice equations defined on a 3x3 stencil
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915016737816576 |
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| author | Hietarinta, Jarmo |
| author_facet | Hietarinta, Jarmo |
| contents | 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 $f_{n,m}=α_{n,m}z+β_{n,m}$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_03582 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Degree growth of lattice equations defined on a 3x3 stencil Hietarinta, Jarmo Exactly Solvable and Integrable Systems 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 $f_{n,m}=α_{n,m}z+β_{n,m}$, 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. |
| title | Degree growth of lattice equations defined on a 3x3 stencil |
| topic | Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2307.03582 |