ANZIAM J. 49 (2007), no. 1, pp. 39–52.

Evolution equations having conservation laws with flux characteristics

B. van Brunt^†	M. Vlieg-Hulstman
Institute of Fundamental Sciences Mathematics Department Massey University New Zealand B.vanBrunt@massey.ac.nz	Institute of Fundamental Sciences Mathematics Department Massey University New Zealand

Received 9 August 2006

Abstract

A class of evolution equations in divergence form is studied in this paper. Specifically, we develop conditions under which the spatial divergence term, the flux, corresponds to the characteristic of a conservation law. The KdV equation is a prominent example of an equation having a flux term that is also a characteristic for a conservation law. We show that the flux term must be self-adjoint. General equations for the corresponding conservation laws and Hamiltonian densities are derived and supplemented with examples.

Download the article in PDF format (size 136 Kb)

2000 Mathematics Subject Classification: primary 35K
(Metadata: XML, RSS, BibTeX)	MathSciNet: MR2378148	Z'blatt-MATH: pre05243890
^†indicates author for correspondence

References

S. C. Anco and G. W. Bluman, “Derivation of conservation laws from nonlocal symmetries of differential equations”, J. Math. Phys. 37 (1996) 2361–2375. MR1385835
S. C. Anco and G. W. Bluman, “Direct construction of conservation laws from field equations”, Phys. Rev. Lett. 78 (1997) 2869–2873. MR1442661
B. Dey, “Domain wall solutions of KdV like equations with higher order nonlinearity”, J. Phys. A: Math. Gen 19 (1986) L9–L12. MR846109
B. Dey, “KdV-like equations with domain wall solutions and their Hamiltonians”, in Solitons: Introduction and Applications: Proceedings Winter School on Solitons, Tiruchirapalli, India, Jan. 1987 (ed. M. Lakshmanan), (Springer-Verlag, New York, 1988), 367–374. MR938122
P. J. Olver, Applications of Lie Groups to Differential Equations (Springer-Verlag, New York, 1986). MR836734
N. Petersson, N. Euler and M. Euler, “Recursion operators for a class of integrable third-order evolution equations”, Stud. Appl. Math. 112 (2004) 201–225. MR2031835
Y. Tan, J. Yang and D. Pelinovsky, “Semi-stability of embedded solitons in the general fifth-order KdV equation”, Wave Motion 36 (2002) 241–255. MR1920719
B. van Brunt, D. Pidgeon, M. Vlieg-Hulstman and W. Halford, “Conservation laws for second-order parabolic partial differential equations”, ANZIAM J. 45 (2004) 333–348. MR2044251

Australian Mathematical Publishing Association Inc.