Cauchy problem for the Ostrovsky equation

Stability and weak rotation limit of solitary waves of the Ostrovsky equation

“…For β < 0 (negative dispersion), the equation models surface and internal waves in the ocean and surface waves in a shallow channel with uneven bottom [2], while for β > 0 (positive dispersion), it models capillary waves on the surface of a liquid and magneto-acoustic waves in a plasma [4,5]. Liu and Varlamov [14] Liu and Varlamov [9] showed using the Concentration Compactness Lemma that there exist solutions of (1.2) in the space X 1 provided c < 2 √ βγ. Moreover, these ground state solutions are characterized as minimizers of I(u; β, c, γ) = βu…”

Hamiltonian formulation, nonintegrability and local bifurcations for the Ostrovsky equation

“…Surprisingly, even if the mass of the solitary wave of the KdV equation is not zero, it is shown [21] that the limit of the solitary waves of the Ostrovsky equation tends to the solitary wave of the KdV equation as the rotation parameter γ tends to zero. For β < 0, solitary waves in the form of stationary localized pulses cannot exist at all [8,20,31]. Equation (3.1) is invariant under the transformation z → −z and thus it is a reversible system.…”

Low-Regularity Solutions for the Ostrovsky Equation

“…Recently, Varlamov and Liu [13] found that the problem (1.1), (1.2) is locally wellposed in the spaceH s (s > 3 2 ) with the condition γ > 0 by using the method of parabolic regularization.…”