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Exceptional hyperbolic systems of Hamiltonian form

Published online by Cambridge University Press:  26 September 2008

P. Carbonaro
P. Carbonaro
Affiliation:
Dipartimento di Matematica dell'Università di Catania, Viale A. Doria 6, 1-95125 Catania, Italy
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Abstract

Some years ago, Bluman [1] gave the ‘integrability conditions’ for a linear second-order hyperbolic Equation, i.e. the conditions under which it can be mapped invertibly to a constant coefficient wave equation. Considering that a second-order hyperbolic equation in two dimensions can be regarded as the hodograph representation of a 2x2 quasi-linear homogeneous system, one may wonder how the fulfilment of the above-mentioned integrability conditions is reflected in the structure of the quasi-linear system. The question is especially interesting if the quasi-linear system derives from a Hamiltonian density.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 6 , Issue 2 , April 1995 , pp. 157 - 167
Copyright
Copyright © Cambridge University Press 1995

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References

[1]Bluman, G. W. 1983 On mapping linear partial differential equations to constant coefficient equations. SIAM J. Appl. Math. 43, 1259.CrossRefGoogle Scholar
[2]Bluman, G. W. & Kumei, S. 1989 Symmetries and Differential Equations. Springer-Verlag.CrossRefGoogle Scholar
[3]Olver, P. J. & Nutku, Y. 1988 Hamiltonian structures for systems of hyperbolic conservation laws. J. Math. Phys. 29, 1610.CrossRefGoogle Scholar
[4]Arik, M., Neyzi, F., Olver, P. J. & Verosky, J. M. 1989 Multi Hamiltonian structure of Born-Infeld equation. J. Math. Phys. 30, 1338.CrossRefGoogle Scholar
[5]Zel'dovich, I. B. & Kompaneets, A. S. 1960 Theory of Detonation. Academic Press.Google Scholar
[6]Jeffrey, A. 1976 Quasilinear hyperbolic systems and waves. Research Notes in Mathematics, Vol. 5. Pitman.Google Scholar
[7]Lax, P. D. 1957 Hyperbolic systems of conservation laws II. Comm. Pure Appl. Math. 10, 537.CrossRefGoogle Scholar
[8]Tsarev, S. P. 1985 On Poisson brackets and one-dimensional Hamiltonian systems of hydrodynamic type. Sov. Math. Dokl. 282, 534.Google Scholar
[9]Carbonaro, P. 1989 A class of completely integrable hyperbolic systems. J. Phys. A: Math. Gen. 22, L667.CrossRefGoogle Scholar
[10]Jeffrey, A. & Donato, A. 1979 The occurrence of singularities in solutions of homogeneous systems of two first order quasilinear hyperbolic equations with smooth initial data. Wave Motion 1, 177–185.CrossRefGoogle Scholar
[11]Landau, L. & Lifchitz, E. 1971 Mécanique des Fluides. Mir, Moscow.Google Scholar