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    FRAENKEL, L. E. 2010. The behaviour near the crest of an extreme Stokes wave. European Journal of Applied Mathematics, Vol. 21, Issue. 02, p. 165.

  • European Journal of Applied Mathematics, Volume 21, Issue 2
  • April 2010, pp. 137-163

On the local uniqueness and the profile of the extreme Stokes wave

  • L. E. FRAENKEL (a1) and P. J. HARWIN (a1)
  • DOI:
  • Published online: 01 February 2010

Proceeding from a recent construction (Fraenkel 2007, Arch. Rational Mech. Anal. 183, 187–214), this paper contains three contributions to the theory of the extreme gravity wave on water (or “wave of greatest height”) initiated by Stokes in 1880. The first is a solution, of the extreme form of the integral equation due to Nekrasov, that consists of an explicit function plus a (rigorously) estimated remainder; the remainder vanishes at the crest and at the trough of the wave and in between has a maximum value less than 0.0026 times the maximum value of the explicit part. The second contribution establishes local uniqueness for a variant of the Nekrasov equation; we prove that this equation has only one solution between upper and lower bounding functions that are moderately far apart. The third contribution consists of a table and graph of strict upper and lower bounds for the interface between air and water in the physical plane.

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[2]C.J. Amick & L.E. Fraenkel (1987) On the behavior near the crest of waves of extreme form. Trans. Amer. Math. Soc. 299, 273298.

[3]C.J. Amick , L.E. Fraenkel & J.F. Toland (1982) On the Stokes conjecture for the wave of extreme form. Acta Math. 148, 193214.

[4]E.D. Cokelet (1977) Steep gravity waves in water of arbitrary uniform depth. Phil. Trans. R. Soc. Lond. A 286, 183230.

[5]L.E. Fraenkel (2007) A constructive existence proof for the extreme Stokes wave. Arch. Rational Mech. Anal. 183, 187214.

[7]I.S. Gandzha & V.P. Lukomsky (2007) On water waves with a corner at the crest. Proc. R. Soc. Lond. A 463, 15971614.

[9]K. Kobayashi (2004) Numerical verification of the global uniqueness of a positive solution for Nekrasov's equation. Japan J. Indust. Appl. Math. 21, 181218.

[10]K. Kobayashi & H. Okamoto (2004) Uniqueness issues on permanent progressive water-waves, J. Nonlinear Math. Phys. 11, 472479.

[11]H. Lewy (1952) A note on harmonic functions and a hydrodynamical application. Proc. Amer. Math. Soc. 3, 111113.

[12]M.S. Longuet-Higgins (1975) Integral properties of periodic gravity waves of finite amplitude. Proc. R. Soc. Lond. A 342, 157174.

[14]J.B. McLeod (1987) The asymptotic behavior near the crest of waves of extreme form. Trans. Amer. Math. Soc. 299, 299302.

[15]R.C.T. Rainey & M.S. Longuet-Higgins (2006) A close one-term approximation to the highest Stokes wave on deep water. Ocean Eng. 33, 20122024.

[17]J.M. Williams (1981) Limiting gravity waves in water of finite depth. Phil. Trans. R. Soc. Lond. A 302, 139188.

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European Journal of Applied Mathematics
  • ISSN: 0956-7925
  • EISSN: 1469-4425
  • URL: /core/journals/european-journal-of-applied-mathematics
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