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.
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Communicated by: Yasuyuki Kawahigashi
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Carpentier, S. A sufficient condition for a rational differential operator to generate an integrable system. Jpn. J. Math. 12, 33–89 (2017). https://doi.org/10.1007/s11537-016-1619-9
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DOI: https://doi.org/10.1007/s11537-016-1619-9
Keywords and phrases
- integrable systems
- Lenard–Magri scheme of integrability
- rational pseudo-differential operators
- symmetries