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

Author(s)
Carpentier, Sylvain
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Abstract
For a rational differential operator L=AB[superscript −1] , the Lenard–Magri scheme of integrability is a sequence of functions F[subscript n],n≥0, such that (1) B(F[subscript n+1])=A(Fn) for all n≥0 and (2) the functions B(F[subscript n]) pairwise commute. We show that, assuming that property (1) holds and that the set of differential orders of B(F[subscript n]) is unbounded, property (2) holds if and only if L belongs to a class of rational operators that we call integrable. If we assume moreover that the rational operator L is 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 (F[subscript n]) starting from any function in Ker B. This result gives some insight in the mechanism of recursion operators, which encode the hierarchies of the corresponding integrable equations.
Date issued
2017-01
URI
http://hdl.handle.net/1721.1/107669
Department
Massachusetts Institute of Technology. Department of Mathematics
Journal
Japanese Journal of Mathematics
Publisher
Springer Japan
Citation
Carpentier, Sylvain. “A Sufficient Condition for a Rational Differential Operator to Generate an Integrable System.” Japanese Journal of Mathematics 12, no. 1 (January 15, 2017): 33–89.
Version: Author's final manuscript
ISSN
0289-2316
1861-3624

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