On the behavior near the crest of waves of extreme form

Authors:
C. J. Amick and L. E. Fraenkel

Journal:
Trans. Amer. Math. Soc. **299** (1987), 273-298

MSC:
Primary 76B15

DOI:
https://doi.org/10.1090/S0002-9947-1987-0869412-4

MathSciNet review:
869412

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Abstract: The angle which the free boundary of an extreme wave makes with the horizontal is the solution of a singular, nonlinear integral equation that does not fit (as far as we know) into the theory of compact operators on Banach spaces. It has been proved only recently that solutions exist and that (as Stokes suggested in 1880) these solutions represent waves with sharp crests of included angle . In this paper we use the integral equation, known properties of solutions and the technique of the Mellin transform to obtain the asymptotic expansion

**8**] that , and follows here that . The derivation of (*) includes an assumption about a question in number theory; if that assumption should be false, logarithmic terms would enter the series at very large values of .

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1987-0869412-4

Keywords:
Water waves,
nonlinear integral equations,
asymptotic analysis

Article copyright:
© Copyright 1987
American Mathematical Society