Capillary Gravity Waves on the Free Surface of an Inviscid Fluid of Infinite Depth. Existence of Solitary Waves

Abstract : Permanent capillary gravity waves on the free surface of a two dimensional inviscid fluid of infinite depth are investigated. An application of the hodograph transform converts the free boundary-value problem into a boundary-value problem for the Cauchy-Riemann equations in the lower halfplane with nonlinear differential boundary conditions. This can be converted to an integro-differential equation with symbol-k^2 + 4|k|-4(1 +µ), where µ is a bifurcation parameter. A normal-form analysis is presented which shows that the boundary-value problem can be reduced to an integrable system of ordinary diflerential equations plus a remainder term containing nonlocal terms of higher order for |µ| small. This normal form system has been studied thoroughly by several authors (Iooss & Kirchgaessner [8], Iooss & Pérouème [10], Dias & Iooss [5]). It admits a pair of solitary-wave solutions which are reversible in the sense of Kirchgaessner [11]. By applying a method introduced in [11], it is shown that this pair of reversible solitary waves persists for the boundary-value problem, and that the decay at infinity of these solitary waves is at least like 1/[x].
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Gérard Iooss, Pius Kirrmann. Capillary Gravity Waves on the Free Surface of an Inviscid Fluid of Infinite Depth. Existence of Solitary Waves . Archive for Rational Mechanics and Analysis, Springer Verlag, 1996, 136 (1), pp.19. ⟨hal-01271137⟩

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