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Nonlinearly resonant surfaces waves and homoclinic bifurcations. (English) Zbl 0671.76019

In this survey paper, recent results are presented concerning the propagation of nonlinear waves on the surface of an inviscid heavy fluid layer of finite depth. In particular, motions of moderate amplitude in steady two-dimensional flows are considered which are induced by a localized or spatially periodic pressure distribution moving along the surface with constant speed. The motion is then described by a quasilinear system of elliptic equations in an infinite cylindrical domain. By nonlinear separation of variables this elliptic system can be reduced to an ordinary differential system of minimal order which has to be investigated. The cases \(b<1/3\) and \(b>1/3\) have to be distinguished, where b is the Bond number \((b=T/\rho gh^ 2\), T surface tension coefficient, \(\rho\) density, g gravity constant, h mean depth of the layer). The cases \(b>1/3\) and \(b=0\) (pure gravity waves) are considered in detail. (In these cases the ordinary differential equation is of second order. For \(0<b<1/3\) a fourth order system is obtained, its analysis is still essentially open.)
Reviewer: W.Müller

MSC:

76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35Q99 Partial differential equations of mathematical physics and other areas of application
34L99 Ordinary differential operators
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