A sufficient condition for a rational differential operator to generate an integrable system
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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.
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Japanese Journal of Mathematics
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.
Author's final manuscript