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Capillary gravity waves on the free surface of an inviscid fluid of infinite depth. Existence of solitary waves. (English) Zbl 0879.76011
Existence of a steadily moving localized solitary wave on a surface of an infinitely deep inviscid fluid is proved. Starting from the fluid mechanics equations, the problem is transformed into a single integro-differential equation resembling the Benjamin-Ono equation. Then, the normal-form analysis reduces the solitary-wave existence problem to an integrable fourth-order dynamical system with a nonlocal residual term. The solitary wave is only weakly localized (similarly to the soliton of the Benjamin-Ono equation), i.e., it decays at infinity nonexponentially. It is proved, nevertheless, that the solution decays at least as \(1/|x|\).

MSC:
76B25 Solitary waves for incompressible inviscid fluids
76B45 Capillarity (surface tension) for incompressible inviscid fluids
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35Q35 PDEs in connection with fluid mechanics
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