|
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
Posted:
April 12, 2007
MathSciNet review:
2318158
Full-text PDF
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.
- 1.
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
(83b:76017), http://dx.doi.org/10.1007/BF00250799
- 2.
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
(83b:76009), http://dx.doi.org/10.1098/rsta.1981.0231
- 3.
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
(83m:35147), http://dx.doi.org/10.1007/BF02392728
- 4.
J.
Thomas Beale, The existence of solitary water waves, Comm.
Pure Appl. Math. 30 (1977), no. 4, 373–389. MR 0445136
(56 #3480)
- 5.
Adrian
Constantin, The trajectories of particles in Stokes waves,
Invent. Math. 166 (2006), no. 3, 523–535. MR 2257390
(2007j:35240), http://dx.doi.org/10.1007/s00222-006-0002-5
- 6.
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
(87f:35210), http://dx.doi.org/10.1080/03605308508820396
- 7.
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 (2003m:76011), http://dx.doi.org/10.1098/rsta.2002.1065
- 8.
Walter
Craig and Peter
Sternberg, Symmetry of solitary waves, Comm. Partial
Differential Equations 13 (1988), no. 5,
603–633. MR
919444 (88m:35132), http://dx.doi.org/10.1080/03605308808820554
- 9.
P.
G. Drazin and R.
S. Johnson, Solitons: an introduction, Cambridge Texts in
Applied Mathematics, Cambridge University Press, Cambridge, 1989. MR 985322
(90j:35166)
- 10.
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
(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.
Ration. Mech. Anal. 171 (2004), no. 3, 349–416.
MR
2038344 (2005f:76017), http://dx.doi.org/10.1007/s00205-003-0292-3
- 13.
J.
J. Stoker, Water waves: The mathematical theory with
applications, Pure and Applied Mathematics, Vol. IV, Interscience
Publishers, Inc., New York, 1957. MR 0103672
(21 #2438)
- 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)
Similar Articles
Retrieve articles in Bulletin of the American Mathematical Society
with MSC (2000):
35J65,
35Q35,
34C05,
76B15
Retrieve articles in all journals
with MSC (2000):
35J65,
35Q35,
34C05,
76B15
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
DOI:
http://dx.doi.org/10.1090/S0273-0979-07-01159-7
PII:
S 0273-0979(07)01159-7
Keywords:
Solitary wave,
potential flow,
particle trajectory.
Received by editor(s):
September 7, 2006
Posted:
April 12, 2007
Article copyright:
© Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
|
