Particle trajectories in solitary water waves
HTML articles powered by AMS MathViewer
- by Adrian Constantin and Joachim Escher PDF
- Bull. Amer. Math. Soc. 44 (2007), 423-431 Request permission
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
- C. J. Amick and J. F. Toland, On solitary water-waves of finite amplitude, Arch. Rational Mech. Anal. 76 (1981), no. 1, 9–95. MR 629699, DOI 10.1007/BF00250799
- C. J. Amick and J. F. Toland, On periodic water-waves and their convergence to solitary waves in the long-wave limit, Philos. Trans. Roy. Soc. London Ser. A 303 (1981), no. 1481, 633–669. MR 647410, DOI 10.1098/rsta.1981.0231
- C. J. Amick, L. E. Fraenkel, and J. F. Toland, On the Stokes conjecture for the wave of extreme form, Acta Math. 148 (1982), 193–214. MR 666110, DOI 10.1007/BF02392728
- J. Thomas Beale, The existence of solitary water waves, Comm. Pure Appl. Math. 30 (1977), no. 4, 373–389. MR 445136, DOI 10.1002/cpa.3160300402
- Adrian Constantin, The trajectories of particles in Stokes waves, Invent. Math. 166 (2006), no. 3, 523–535. MR 2257390, DOI 10.1007/s00222-006-0002-5
- Walter Craig, An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits, Comm. Partial Differential Equations 10 (1985), no. 8, 787–1003. MR 795808, DOI 10.1080/03605308508820396
- Walter Craig, Non-existence of solitary water waves in three dimensions, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 360 (2002), no. 1799, 2127–2135. Recent developments in the mathematical theory of water waves (Oberwolfach, 2001). MR 1949966, DOI 10.1098/rsta.2002.1065
- Walter Craig and Peter Sternberg, Symmetry of solitary waves, Comm. Partial Differential Equations 13 (1988), no. 5, 603–633. MR 919444, DOI 10.1080/03605308808820554
- P. G. Drazin and R. S. Johnson, Solitons: an introduction, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1989. MR 985322, DOI 10.1017/CBO9781139172059
- L. E. Fraenkel, An introduction to maximum principles and symmetry in elliptic problems, Cambridge Tracts in Mathematics, vol. 128, Cambridge University Press, Cambridge, 2000. MR 1751289, DOI 10.1017/CBO9780511569203
- K. O. Friedrichs and D. H. Hyers, The existence of solitary waves, Comm. Pure Appl. Math. 7 (1954), 517–550. MR 65317, DOI 10.1002/cpa.3160070305
- P. I. Plotnikov and J. F. Toland, Convexity of Stokes waves of extreme form, Arch. Ration. Mech. Anal. 171 (2004), no. 3, 349–416. MR 2038344, DOI 10.1007/s00205-003-0292-3
- J. J. Stoker, Water waves: The mathematical theory with applications, Pure and Applied Mathematics, Vol. IV, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1957. MR 0103672
Additional Information
- Adrian Constantin
- Affiliation: School of Mathematics, Trinity College Dublin, Dublin 2, Ireland; and Department of Mathematics, Lund University, 22100 Lund, Sweden
- Email: adrian@maths.tcd.ie, adrian.constantin@math.lu.se
- Joachim Escher
- Affiliation: Institut für Angewandte Mathematik, Leibniz Universität Hannover, Welfengarten 1 30167 Hannover, Germany
- Email: escher@ifam.uni-hannover.de
- Received by editor(s): September 7, 2006
- Published electronically: April 12, 2007
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Bull. Amer. Math. Soc. 44 (2007), 423-431
- MSC (2000): Primary 35J65, 35Q35, 34C05, 76B15
- DOI: https://doi.org/10.1090/S0273-0979-07-01159-7
- MathSciNet review: 2318158