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Wave Motion: Theory, Modelling, and Computation pp 295–336Cite as

The Small Dispersion Limit of the Korteweg-De Vries Equation

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Part of the book series: Mathematical Sciences Research Institute Publications ((MSRI,volume 7))

Abstract

There are many physical systems which display shocks i.e. regions in space where the solution develops extremely large slopes. In general, such systems are too complicated to be treated by exact calculation and their properties are best studied through the proof of general theorems. A model of the formation and propagation of dispersive shocks in one space dimension, in which explicit calculation is possible, is given by the initial value problem for the Korteweg-de Vries equation:

$$ {u_{t}} - 6u{u_{x}} + {\varepsilon ^{2}}u{u_{{xxx}}} = 0 $$
((1.1a))
$$ u(x,o,\varepsilon ) = - v(x)$$
((1.1b))

in the limit ε → 0.

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Author information

Author notes

  1. Stephanos Venakides

    Present address: Mathematics, Duke University, Durham, NC, 27706, USA

Authors and Affiliations

  1. Stanford University, USA

    Stephanos Venakides

Authors
  1. Stephanos Venakides
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Editor information

Editors and Affiliations

  1. Department of Mathematics, University of California, Berkeley, 94720, California, USA

    Alexandre J. Chorin

  2. Department of Mathematics, Princeton University, Princeton, New Jersey, 08544, USA

    Andrew J. Majda

Additional information

To Peter Lax on his 60th birthday.

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© 1987 Springer-Verlag New York Inc.

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Venakides, S. (1987). The Small Dispersion Limit of the Korteweg-De Vries Equation. In: Chorin, A.J., Majda, A.J. (eds) Wave Motion: Theory, Modelling, and Computation. Mathematical Sciences Research Institute Publications, vol 7. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9583-6_12

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  • DOI: https://doi.org/10.1007/978-1-4613-9583-6_12

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4613-9585-0

  • Online ISBN: 978-1-4613-9583-6

  • eBook Packages: Springer Book Archive

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