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

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.