Searching for integrable lattice maps using factorization.pdf

Searching for integrable lattice maps using factorization Jarmo Hietarinta Department of Physics, University of Turku FIN–20014 Turku, Finland 7 0 0 Claude Viallet 2 n UMR 7589 Centre National de la Recherche Scientifique u Universit´e Paris VI, Universit´e Paris VII J 8 Boˆıte 126, 4 place Jussieu ] F–75252 Paris Cedex 05, France I S . n February 1, 2008 i l n [ 2 v Abstract 3 0 We analyze the factorization process for lattice maps, searching for integrable cases. 9 1 The maps were assumed to be at most quadratic in the dependent variables, and we 5. required minimal factorization (one linear factor) after 2 steps of iteration. The results 0 were then classified using algebraic entropy. Some new models with polynomial growth 7 (strongly associated with integrability) were found. One of them is a nonsymmetric 0 : generalization of the homogeneous quadratic maps associated with KdV (modified and v Schwarzian), for this new model we have also verified the “consistency around a cube”. i X r a 1 Introduction Although many interesting results have been obtained for discrete integrable systems and many properties have been clarified, we are still to great extent in the “taxonomic” stage of development. We do not have any clear classification and probably we have only discovered a small sample of integrable difference equations, the tip of the iceberg. As with differential equations, there is no universal definition of integrability for discrete systems,