A sufficient condition for a rational differential operator to generate an integrable system
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 (...
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| Format: | Article |
| Language: | English |
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Springer Japan
2017
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| Online Access: | http://hdl.handle.net/1721.1/107669 https://orcid.org/0000-0001-6809-4128 |
| _version_ | 1826207253070348288 |
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| author | Carpentier, Sylvain |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Carpentier, Sylvain |
| author_sort | Carpentier, Sylvain |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T13:46:26Z |
| format | Article |
| id | mit-1721.1/107669 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T13:46:26Z |
| publishDate | 2017 |
| publisher | Springer Japan |
| record_format | dspace |
| spelling | mit-1721.1/1076692019-05-17T08:43:47Z A sufficient condition for a rational differential operator to generate an integrable system Carpentier, Sylvain Massachusetts Institute of Technology. Department of Mathematics Carpentier, Sylvain 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. 2017-03-23T18:28:07Z 2017-11-05T05:00:05Z 2017-01 2016-05 2017-03-03T04:48:33Z Article http://purl.org/eprint/type/JournalArticle 0289-2316 1861-3624 http://hdl.handle.net/1721.1/107669 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. PUBLISHER_POLICY https://orcid.org/0000-0001-6809-4128 en http://dx.doi.org/10.1007/s11537-016-1619-9 Japanese Journal of Mathematics Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. The Mathematical Society of Japan and Springer Japan text/xml Springer Japan Springer Japan |
| spellingShingle | Carpentier, Sylvain A sufficient condition for a rational differential operator to generate an integrable system |
| title | A sufficient condition for a rational differential operator to generate an integrable system |
| title_full | A sufficient condition for a rational differential operator to generate an integrable system |
| title_fullStr | A sufficient condition for a rational differential operator to generate an integrable system |
| title_full_unstemmed | A sufficient condition for a rational differential operator to generate an integrable system |
| title_short | A sufficient condition for a rational differential operator to generate an integrable system |
| title_sort | sufficient condition for a rational differential operator to generate an integrable system |
| url | http://hdl.handle.net/1721.1/107669 https://orcid.org/0000-0001-6809-4128 |
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