Conservation laws for a class of third order evolutionary differential systems

Author:
Sung Ho Wang

Translated by:

Journal:
Trans. Amer. Math. Soc. **356** (2004), 4055-4073

MSC (2000):
Primary 35K25; Secondary 58A15

DOI:
https://doi.org/10.1090/S0002-9947-04-03501-9

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 , , . 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

Email:
wang@postech.ac.kr

DOI:
https://doi.org/10.1090/S0002-9947-04-03501-9

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