A sufficient condition for a rational differential operator to generate an integrable system |
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Authors: | Sylvain Carpentier |
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Institution: | 1.Department of Mathematics,Massachusetts Institute of Technology,Cambridge,USA |
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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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