ISSN:
1572-9036
Keywords:
58G37
;
35Q53
;
17B81
;
PDE
;
symmetries
;
KdV
;
EW
;
jet bundles
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract Using a definition of the (infinite-dimensional) Lie algebras introduced by Estabrook and Wahlquist (EW) that ‘lives’ directly on the jet bundle, we establish explicit methods to determine prolongations of these algebras to higher jet spaces, and calculate these prolongations for Burgers' equation and the KdV equation, to arbitrary level on the infinite jet bundle. Each level of prolongation introduces additional generator(s) of the algebra. For Burgers' equation, we first find the Kac-Moody algebra that is the (universal) completion of that algebra on the lowest-order jet space. We then show how each new generator creates a free flow of the lower-order algebra over the jet bundle, without requiring any new commutation relations. For the KdV equation, one new generator creates a flow, but the other introduces into the structure of the prolonged total derivative operators the characteristics for the hierarchy of conserved quantities of the PDE, thereby mixing together these ‘nonlocal’ symmetries with the local ones in a way that may be valuable. In the limit to the infinite jet bundle, we acquire the EW algebra for the entire KdV hierarchy of PDE's, moving toward a comparison of this approach with the work of the Japanese school.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01082449
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