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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Conservation laws for a class of third order evolutionary differential systems

Author: Sung Ho Wang
Journal: Trans. Amer. Math. Soc. 356 (2004), 4055-4073
MSC (2000): Primary 35K25; Secondary 58A15
Published electronically: February 27, 2004
MathSciNet review: 2058518
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Abstract: Conservation laws of third order quasi-linear scalar evolution equations are studied via exterior differential system and characteristic cohomology. We find a subspace of 2-forms in the infinite prolongation space in which every conservation law has a unique representative. Analysis of the structure of this subspace based upon the symbol of the differential equation leads to a universal integrability condition for an evolution equation to admit any higher order (weight) conservation laws. As an example, we give a complete classification of a class of evolution equations which admit conservation laws of the first three consecutive weights $-1$, $1$, $3$. The differential system describing the flow of a curve in the plane by the derivative of its curvature with respect to the arc length is also shown to exhibit the KdV property, i.e., an infinite sequence of conservation laws of distinct weights.

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Additional Information

Sung Ho Wang
Affiliation: Department of Mathematics, Postech, Pohang, Korea 790-784

Keywords: Third order scalar evolution equation, exterior differential system, conservation law, characteristic cohomology
Received by editor(s): June 9, 2003
Received by editor(s) in revised form: July 8, 2003
Published electronically: February 27, 2004
Article copyright: © Copyright 2004 American Mathematical Society