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A sufficient condition for a rational differential operator to generate an integrable system

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Japan. J. Math. 12, 33–89 (2017) DOI: 10.1007/s11537-016-1619-9

A sufficient condition for a rational differential operator to generate an integrable system

Sylvain Carpentier

Received: 20 May 2016 / Revised: 19 October 2016 / Accepted: 1 November 2016 Published online: 15 January 2017

© The Mathematical Society of Japan and Springer Japan 2017 Communicated by: Yasuyuki Kawahigashi

Abstract. For a rational differential operatorLDAB1, the Lenard–Magri scheme of integra- bility is a sequence of functionsFn; n0, such that (1)B.FnC1/DA.Fn/for alln0and (2) the functionsB.Fn/pairwise commute. We show that, assuming that property.1/holds and that the set of differential orders ofB.Fn/is unbounded, property.2/holds if and only ifLbelongs to a class of rational operators that we call integrable. If we assume moreover that the rational operatorLis weakly non-local and preserves a certain splitting of the algebra of functions into even and odd parts, we show that one can always find such a sequence .Fn/starting from any function in KerB. This result gives some insight in the mechanism of recursion operators, which encode the hierarchies of the corresponding integrable equations.

Keywords and phrases:integrable systems, Lenard–Magri scheme of integrability, rational pseudo- differential operators, symmetries

Mathematics Subject Classification (2010): 37K10, 17B80, 35Q53, 35S05, 37K10

S. CARPENTIER

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

(e-mail:[email protected])

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