A sufficient condition for a Rational Differential Operator to generate an Integrable System

TL;DR

This paper characterizes when a rational differential operator can generate integrable systems via the Lenard-Magri scheme, providing conditions under which the scheme's properties are equivalent and exploring the structure of such operators.

Contribution

It introduces the class of integrable rational operators and establishes criteria for their role in generating integrable hierarchies, advancing understanding of recursion operators.

Findings

Property (2) holds iff L is integrable under unbounded order assumption.

Weakly non-local operators with certain symmetries always generate Lenard-Magri sequences.

The results clarify the structure of recursion operators in integrable systems.

Abstract

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 $(F_n)$ 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.