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Instability and collapse of waveguides on the water surface under the ice cover. (English) Zbl 1119.76333
Summary: We consider long gravity-flexural waves on a surface of a perfect liquid of a finite depth under elastic ice-plate. Such waves of small but finite amplitude are governed by the generalized Kadomtsev-Petviashvili (KP) equation, which contains higher spatial derivatives. The generalized KP equation admits waveguide solutions, describing waves periodic in the direction of propagation and localized in the transverse direction. Waveguide represents a wave being the nonlinear product of instability of carrier monochromatic wave with respect to transverse perturbations. Instability of waveguides in its nonlinear stage is studied. For this purpose we use the alternative description via the Davey-Stewartson (DS) equations for slowly varying amplitudes of monochromatic waves. The DS equations are asymptotically equivalent to the initial generalized KP equation. Behaviour of perturbations is determined by values of wavenumber of the carrier wave. We find, in particular, that for some range of wavenumbers of the carrier wave the waveguide is subjected to the local collapse which differs from collapse of waveguides in the fluid of infinite depth.

MSC:
76E30 Nonlinear effects in hydrodynamic stability
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35Q53 KdV equations (Korteweg-de Vries equations)
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