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Particle trajectories in solitary water waves

Authors: Adrian Constantin and Joachim Escher
Journal: Bull. Amer. Math. Soc. 44 (2007), 423-431
MSC (2000): Primary 35J65, 35Q35, 34C05, 76B15
Published electronically: April 12, 2007
MathSciNet review: 2318158
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Abstract | References | Similar Articles | Additional Information

Abstract: Analyzing a free boundary problem for harmonic functions in an infinite planar domain, we prove that in a solitary water wave each particle is transported in the wave direction but slower than the wave speed. As the solitary wave propagates, all particles located ahead of the wave crest are lifted, while those behind it experience a downward motion, with the particle trajectory having asymptotically the same height above the flat bed.

References [Enhancements On Off] (What's this?)

  • 1. C. J. Amick and J. F. Toland, On solitary waves of finite amplitude, Arch. Rat. Mech. Anal. 76 (1981), 9-95. MR 629699 (83b:76017)
  • 2. C. J. Amick and J. F. Toland, On periodic water waves and their convergence to solitary waves in the long-wave limit, Phil. Trans. Roy. Soc. London 303 (1981), 633-673. MR 647410 (83b:76009)
  • 3. C. J. Amick, L. E. Fraenkel and J. F. Toland, On the Stokes conjecture for the wave of extreme form, Acta Mathematica 148 (1982), 193-214. MR 666110 (83m:35147)
  • 4. J. T. Beale, The existence of solitary water waves, Comm. Pure Appl. Math. 30 (1977), 373-389. MR 0445136 (56:3480)
  • 5. A. Constantin, The trajectories of particles in Stokes waves, Invent. Math. 166 (2006), 523-535. MR 2257390
  • 6. W. Craig, An existence theory for water waves, and Boussinesq and Korteweg-de Vries scaling limits, Comm. PDE 10 (1985), 787-1003. MR 795808 (87f:35210)
  • 7. W. Craig, Nonexistence of solitary water waves in three dimensions, Phil. Trans. Royal Soc. London A 360 (2002), 1-9. MR 1949966 (2003m:76011)
  • 8. W. Craig and P. Sternberg, Symmetry of solitary waves, Comm. PDE 13 (1988), 603-633. MR 919444 (88m:35132)
  • 9. P. G. Drazin and R. S. Johnson, Solitons: An Introduction, Cambridge University Press, Cambridge, 1989. MR 985322 (90j:35166)
  • 10. L. E. Fraenkel, An Introduction to Maximum Principles and Symmetry in Elliptic Problems, Cambridge University Press, Cambridge, 2000. MR 1751289 (2001c:35042)
  • 11. K. O. Friedrichs and D. H. Hyers, The existence of solitary waves, Comm. Pure Appl. Math. 7 (1954), 517-550. MR 0065317 (16:413f)
  • 12. P. I. Plotnikov and J. F. Toland, Convexity of Stokes waves of extreme form, Arch. Rat. Mech. Anal. 171 (2004), 349-416. MR 2038344 (2005f:76017)
  • 13. J. J. Stoker, Water Waves, Interscience Publ. Inc., New York, 1957. MR 0103672 (21:2438)

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Additional Information

Adrian Constantin
Affiliation: School of Mathematics, Trinity College Dublin, Dublin 2, Ireland; and Department of Mathematics, Lund University, 22100 Lund, Sweden

Joachim Escher
Affiliation: Institut für Angewandte Mathematik, Leibniz Universität Hannover, Welfengarten 1 30167 Hannover, Germany

Keywords: Solitary wave, potential flow, particle trajectory.
Received by editor(s): September 7, 2006
Published electronically: April 12, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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