RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Regul. Chaotic Dyn.: Year: Volume: Issue: Page: Find

 Regul. Chaotic Dyn., 2017, Volume 22, Issue 4, Pages 386–497 (Mi rcd262)

Rational Integrability of Trigonometric Polynomial Potentials on the Flat Torus

Thierry Combot

Scuola Normale Superiore, Centro di Ricerca Matematica Ennio De Giorgi, Laboratorio Fibonacci, Piazza Cavalieri, 56127 Pisa

Abstract: We consider a lattice $\mathcal{L}\subset \mathbb{R}^n$ and a trigonometric potential $V$ with frequencies $k\in\mathcal{L}$. We then prove a strong rational integrability condition on $V$, using the support of its Fourier transform. We then use this condition to prove that a real trigonometric polynomial potential is rationally integrable if and only if it separates up to rotation of the coordinates. Removing the real condition, we also make a classification of rationally integrable potentials in dimensions $2$ and $3$ and recover several integrable cases. After a complex change of variables, these potentials become real and correspond to generalized Toda integrable potentials. Moreover, along the proof, some of them with high-degree first integrals are explicitly integrated.

Keywords: trigonometric polynomials, differential Galois theory, integrability, Toda lattice

DOI: https://doi.org/10.1134/S1560354717040049

References: PDF file   HTML file

Bibliographic databases:

MSC: 37J30
Accepted:01.06.2017
Language:

Citation: Thierry Combot, “Rational Integrability of Trigonometric Polynomial Potentials on the Flat Torus”, Regul. Chaotic Dyn., 22:4 (2017), 386–497

Citation in format AMSBIB
\Bibitem{Com17} \by Thierry Combot \paper Rational Integrability of Trigonometric Polynomial Potentials on the Flat Torus \jour Regul. Chaotic Dyn. \yr 2017 \vol 22 \issue 4 \pages 386--497 \mathnet{http://mi.mathnet.ru/rcd262} \crossref{https://doi.org/10.1134/S1560354717040049} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000407398500004} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85026864525}