Abstract
Chen & Saffman [4] have given a weakly nonlinear theory of steady periodic two-dimensional irrotational waves on the surface of a perfect fluid of infinite depth which takes account of the effects of gravity and surface tension. They remark that the question of existence of such waves is “a non-trivial and still incompletely solved problem”. It is our purpose in the present paper to give a rigorous mathematical account of the existence theory for such waves of small amplitude. In so doing we are able to give a firm basis for the more formal aspects of Chen & Saffman’s work, and to vindicate many of their main conclusions. Our work will be couched throughout in the language of modern bifurcation theory, and is a consequence of the Lyapunov-Schmidt reduction procedure in the presence of certain symmetry considerations. Other studies have contributed to the rigorous theory, but it is only possible to comment further once the problem has been described in detail.
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This Paper is Dedicated to Professor James Serrin on the Occasion of his Sixtieth Birthday
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© 1989 Springer-Verlag Berlin Heidelberg
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Jones, M., Toland, J. (1989). Symmetry and the Bifurcation of Capillary-Gravity Waves. In: Analysis and Continuum Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-83743-2_18
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DOI: https://doi.org/10.1007/978-3-642-83743-2_18
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