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Self-channelling of surface water waves in the presence of an additional surface pressure. (English) Zbl 0942.76020
Summary: We examine the nonlinear instability of long periodic waves of a small amplitude on a surface of a water layer of finite depth either subjected to surface tension or in the presence of an elastic ice-sheet floating on the water surface. Wave processes in both cases are described by a model equation which generalizes the Kadomtsev-Petviashvili equation to the presence of higher-order dispersive effects. The treatment is based on the analysis of the Benjamin-Feir type instability, governed by the Davey-Stewartson equations for slowly varying in time and space complex amplitudes of periodic waves. The homogeneous periodic wave is shown to be unstable under perturbations transversal to the direction of wave propagation. Such kind of instability leads to a formation of a lattice of essential wave-guides, i.e. waves periodic in the direction of propagation and localized in the transversal direction. We discuss some natural effects of ice damage, which can be explained with the help of such an instability.

##### MSC:
 76E17 Interfacial stability and instability in hydrodynamic stability 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction 76E30 Nonlinear effects in hydrodynamic stability 86A40 Glaciology
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##### References:
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