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Interacting Weakly Nonlinear Hyperbolic and Dispersive Waves

  • Conference paper
Microlocal Analysis and Nonlinear Waves

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 30))

Abstract

Weakly nonlinear wave interactions are resonant or nonresonant. The linearized dispersion relation of the wave motion determines the resonant interactions. Resonant interactions cause significant changes in the wave-field. The evolution of the wave-field is determined using weakly nonlinear asymptotics. Quadratically nonlinear resonant interactions of dispersive waves satisfy the three wave resonance condition. The wave amplitudes solve the three wave resonant interaction equations. The phase velocity of hyperbolic waves is independent of frequency. As a result, hyperbolic waves participate in many resonant interactions. The amplitude of a single hyperbolic wave satisfies the inviscid Burgers equation. Harmonic resonance causes wave-form distortion and shock formation. The amplitudes of several interacting hyperbolic waves solve a system of integro-differential equations. The interaction of three oblique hyperbolic planar waves can generate a countably infinite family of new waves. Weak resonance of nonplanar hyperbolic waves also generates infinitely many new waves.

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Author information

Author notes

  1. John K. Hunter

    Present address: Department of Mathematics, University of California, Davis, CA, 95616, USA

Authors and Affiliations

  1. Department of Mathematics, Colorado State University, Fort Collins, CO, 80523, USA

    John K. Hunter

Authors
  1. John K. Hunter
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Editor information

Editors and Affiliations

  1. Department of Mathematics, Rutgers University, New Brunswick, NJ, 08903, USA

    Michael Beals

  2. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Richard B. Melrose

  3. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109-1003, USA

    Jeffrey Rauch

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© 1991 Springer-Verlag New York, Inc.

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Hunter, J.K. (1991). Interacting Weakly Nonlinear Hyperbolic and Dispersive Waves. In: Beals, M., Melrose, R.B., Rauch, J. (eds) Microlocal Analysis and Nonlinear Waves. The IMA Volumes in Mathematics and its Applications, vol 30. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9136-4_7

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  • DOI: https://doi.org/10.1007/978-1-4613-9136-4_7

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4613-9138-8

  • Online ISBN: 978-1-4613-9136-4

  • eBook Packages: Springer Book Archive

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