Particle trajectories in solitary water waves

Constantin, Adrian; Escher, Joachim

doi:10.1090/S0273-0979-07-01159-7

Particle trajectories in solitary water waves
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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