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A differential equation in the theory of resonant oscillations of water waves

Published online by Cambridge University Press:  14 November 2011

C. J. Amick  and
J. F. Toland
C. J. Amick
Affiliation:
Department of Mathematics, University of Chicago, 5734 University Avenue, Chicago, Illinois 60637, U.S.A.
J. F. Toland
Affiliation:
School of Mathematical Sciences, University of Bath, Bath BA2 7AY, U.K.
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Synopsis

We show that the problem

has a unique solution and that the asymptotic behaviour of this solution as x → −∞ is given by w(x) = x + B ∣x∣−1/4 cos (b ∣x∣3/2 + φ) + O(x−1), for some constant φ. Here B ≠ 0 depends on ε and In particular there are no even solutions of problem (*) for any ε > 0.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 114 , Issue 1-2 , 1990 , pp. 15 - 26
Copyright
Copyright © Royal Society of Edinburgh 1990

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