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

Iooss, Gérard ; Kirrmann, Pius

HAL, hal-01271137 / Harvested from HAL

Résumé

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].

Détail

Publié le : 1996-07-04
Classification:  normal forms,  solitary waves in infinite depth,  water waves,  continuous spectrum,  reversible dynamics,  [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP],  [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Mechanics of the fluids [physics.class-ph],  [NLIN.NLIN-PS]Nonlinear Sciences [physics]/Pattern Formation and Solitons [nlin.PS],  [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS]