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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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References

1Byatt-Smith, J. G. B.. Resonant oscillations in shallow water with small mean-square disturbances. J. Fluid Mech. 193 (1988), 369–390.CrossRefGoogle Scholar
2Byatt-Smith, J. G. B.. On the solutions of a second order differential equation arising in the theory of resonant oscillations in a tank. Stud. Appl. Math. 179 (1988), 143–157.CrossRefGoogle Scholar
3Byatt-Smith, J. G. B.. The asymptotic solution of a connection problem of a second order differential equation (preprint).Google Scholar
4Ockendon, H., Ockendon, J. R. and Johnson, A. D.. Resonant sloshing in shallow water. J. Fluid Mech. 167 (1986), 465–479.CrossRefGoogle Scholar
5Langer, J. S.. Existence of needle crystals in local models of solidification. Phys. Rev. A33 (1986), 435–441.CrossRefGoogle Scholar
6Kruskal, M. D. and Segur, H.. Asymptotics beyond all order in a model of crystal growth (preprint).Google Scholar
7Amick, C. J. and McLeod, J. B.. A singular perturbation problem in needle crystals. Arch. Rational Mech. Anal. 109 (1990), 139–171.CrossRefGoogle Scholar
8Caratheodory, C.. Theory of Functions of a Complex Variable, Vol. 1 (New York: Chelsea, 1984).Google Scholar
9Gilbarg, D. and Trudinger, N. S.. Elliptic Partial Differential Equations of Second Order 2nd Ed (Berlin: Springer, 1983).Google Scholar