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Multidimensional Hyperbolic Problems and Computations pp 281–310Cite as

An Introduction to Weakly Nonlinear Geometrical Optics

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Part of the book series: The IMA Volumes in Mathematics and Its Applications ((IMA,volume 29))

Abstract

Many natural phenomenae are governed by systems of nonlinear conservation laws that are — in a first approximation — hyperbolic. In this context, the understanding of the laws governing the propagation and interaction of small but finite amplitude high frequency waves in hyperbolic P.D.E.’s, and their interactions with “large scale” phenomenae (mean flows, shear layers, shock and detonation waves, etc.) is very important. Weakly Nonlinear Geometrical Optics (W.N.G.O.) is an asymptotic formal theory whose objective is precisely to do this.

In this paper we review the theory of W.N.G.O. as it stands currently. An important point is that W.N.G.O. is not (yet?) a complete theory. It fails at caustics, singular rays, etc. Since these are frequently regions of physical interest, a good theory for what happens there is important — and mostly nonexistent. The current status of these important open problems will also be (briefly) reviewed.

This work was partially supported by grant NSF #DMS-8702625 and was performed, in part, while the author visited the IMA during April 1989.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, Room 2-337, Cambridge, MA, 02139, USA

    Rodolfo R. Rosales

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  1. Rodolfo R. Rosales
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Editors and Affiliations

  1. Department of Applied Mathematics and Statistics, SUNY at Stony Brook, Stony Brook, NY, 11794-3600, USA

    James Glimm

  2. Department of Mathematics and Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ, 08544, USA

    Andrew J. Majda

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© 1991 Springer-Verlag New York, Inc.

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Rosales, R.R. (1991). An Introduction to Weakly Nonlinear Geometrical Optics. In: Glimm, J., Majda, A.J. (eds) Multidimensional Hyperbolic Problems and Computations. The IMA Volumes in Mathematics and Its Applications, vol 29. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9121-0_22

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  • DOI: https://doi.org/10.1007/978-1-4613-9121-0_22

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4613-9123-4

  • Online ISBN: 978-1-4613-9121-0

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