American Institute of Mathematical Sciences

December  2019, 6(2): 277-306. doi: 10.3934/jcd.2019014

Integrable reductions of the dressing chain

 1 Department of Mathematics, Faculty of Science, University of Hradec Kralove, Czech Republic 2 Department of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus 3 Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR 7348 du CNRS, Bât. H3, Boulevard Marie et Pierre Curie, Site du Futuroscope, TSA 61125, 86073 POITIERS Cedex 9, France

* Corresponding author: Charalampos Evripidou

Received  March 2019 Revised  July 2019 Published  November 2019

Fund Project: The research of the first author was supported by the project "International mobilities for research activities of the University of Hradec Králové", CZ.02.2.69/0.0/0.0/16_027/0008487.

In this paper we construct a family of integrable reductions of the dressing chain, described in its Lotka-Volterra form. For each $k, n\in \mathbb N$ with $n \geqslant 2k+1$ we obtain a Lotka-Volterra system $\hbox{LV}_b(n, k)$ on $\mathbb {R}^n$ which is a deformation of the Lotka-Volterra system $\hbox{LV}(n, k)$, which is itself an integrable reduction of the $2m+1$-dimensional Bogoyavlenskij-Itoh system $\hbox{LV}({2m+1}, m)$, where $m = n-k-1$. We prove that $\hbox{LV}_b(n, k)$ is both Liouville and non-commutative integrable, with rational first integrals which are deformations of the rational first integrals of $\hbox{LV}({n}, {k})$. We also construct a family of discretizations of $\hbox{LV}_b(n, 0)$, including its Kahan discretization, and we show that these discretizations are also Liouville and superintegrable.

Citation: Charalampos Evripidou, Pavlos Kassotakis, Pol Vanhaecke. Integrable reductions of the dressing chain. Journal of Computational Dynamics, 2019, 6 (2) : 277-306. doi: 10.3934/jcd.2019014
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