Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2017 / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 129-133). / A key feature of integrability for systems of evolution PDEs ut = F(u), where F lies in a differential algebra of functionals V and u = (U1, ... , ul) depends on one space variable x and time t, is to be part of an infinite hierarchy of generalized symmetries. Recall that V carries a Lie algebra bracket {F, G} = XF(G) - XG(F), where XF denotes the evolutionnary vector field attached to F. In all known examples, these hierarchies are constructed by means of Lenard-Magri sequences: one can find a pair of matrix differential operators (A(a), B(a)) and a sequence (G.n)>n>0,[epsilon] Vl such that ** F = B(GN) for some N >/= 0, ** {B(Gn), B(Gm)} = 0 for all n, m >/= 0, ** B(G,+1 ) = A(G) for all n,m >/= 0. We show that in the scalar case l = 1 a necessary condition for a pair of differential operators (A, B) to generate a Lenard-Magri sequence is that for all constants [lambda], the family C[lambda] = A + [lambda]B must satisfy for all F, G [epsilon]V {C[lambda](F), C[lambda](G)} [epsilon] ImC[lambda]. We call such pairs integrable. We give a sufficient condition on an integrable pair of matrix differential operators (A, B) to generate an infinite Lenard- Magri sequence when the rational matrix differential operator L = AB-1 is weakly non-local and the algebra of differential functions V is either Z or Z/2Z-graded. This is applied to many systems of evolution PDEs to prove their integrability. / by Sylvain Carpentier. / Ph. D. / Ph.D. Massachusetts Institute of Technology, Department of Mathematics
| Identifer | oai:union.ndltd.org:MIT/oai:dspace.mit.edu:1721.1/112909 |
| Date | January 2017 |
| Creators | Carpentier, Sylvain,Ph. D.Massachusetts Institute of Technology. |
| Contributors | Victor G. Kac., Massachusetts Institute of Technology. Department of Mathematics., Massachusetts Institute of Technology. Department of Mathematics |
| Publisher | Massachusetts Institute of Technology |
| Source Sets | M.I.T. Theses and Dissertation |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | 133 pages, application/pdf |
| Rights | MIT theses are protected by copyright. They may be viewed, downloaded, or printed from this source but further reproduction or distribution in any format is prohibited without written permission., http://dspace.mit.edu/handle/1721.1/7582 |
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