2026-09-19T12:59:32Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/1129092026-06-16T18:55:36Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022
Victor G. Kac.
Carpentier, Sylvain,Ph. D.Massachusetts Institute of Technology.
Massachusetts Institute of Technology. Department of Mathematics.
Massachusetts Institute of Technology. Department of Mathematics
2017-12-20T18:16:56Z
2017-12-20T18:16:56Z
2017
2017
http://hdl.handle.net/1721.1/112909
1015202848
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
133 pages
eng
Massachusetts Institute of Technology
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
Mathematics.
Rational matrix differential operators and integrable systems of PDEs
Thesis
application/pdf
2019-06-17T20:30:16Z
Publication
unknown
unknown
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<Abstract>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.</Abstract>
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