Japanese Journal of Mathematics

, Volume 12, Issue 1, pp 33–89

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

Original Paper

DOI: 10.1007/s11537-016-1619-9

Cite this article as:
Carpentier, S. Jpn. J. Math. (2017) 12: 33. doi:10.1007/s11537-016-1619-9


For a rational differential operator \({L=AB^{-1}}\), the Lenard–Magri scheme of integrability is a sequence of functions \({F_n, n \geq 0}\), such that (1) \({B(F_{n+1})=A(F_n)}\) for all \({n \geq 0}\) and (2) the functions \({B(F_n)}\) pairwise commute. We show that, assuming that property (1) holds and that the set of differential orders of \({B(F_n)}\) 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 (Fn) 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.

Keywords and phrases

integrable systems Lenard–Magri scheme of integrability rational pseudo-differential operators symmetries 

Mathematics Subject Classification (2010)

37K10 17B80 35Q53 35S05 37K10 

Copyright information

© The Mathematical Society of Japan and Springer Japan 2017

Authors and Affiliations

  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA