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Long-time solutions of the Ostrovsky equation. (English) Zbl 1194.35366

Summary: The Ostrovsky equation is a modification of the Korteweg-de Vries equation which takes account of the effects of background rotation. It is well known that the usual Korteweg-de Vries solitary wave decays and is replaced by radiating inertia gravity waves. Here we show through numerical simulations that after a long-time a localized wave packet emerges as a persistent and dominant feature. The wavenumber of the carrier wave is associated with that critical wavenumber where the underlying group velocity is a minimum (in absolute value). Based on this feature, we construct a weakly nonlinear theory leading to a higher-order nonlinear Schrödinger equations in an attempt to describe the numerically found wave packets.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35Q35 PDEs in connection with fluid mechanics
35Q55 NLS equations (nonlinear Schrödinger equations)
35B40 Asymptotic behavior of solutions to PDEs
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76E07 Rotation in hydrodynamic stability
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References:

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