Sharp well-posedness of the Cauchy problem for a generalized Ostrovsky equation with positive dispersion

Here u(x, t) represents the free surface of the liquid and the parameter γ > 0 measures the effect of rotation. (1.1) describes the propagation of internal waves of even modes in the ocean; for instance, see the work of Galkin and Stepanyants [1], Leonov [2], and Shrira [3, 4]. The parameter β determines the type of dispersion, more precisely, when β < 0, (1.1) denotes the generalized Ostrovsky equation with negative dispersion; when β > 0, (1.1) denotes the generalized Ostrovsky equation with positive dispersion. When γ = 0, (1.1) reduces to the modified Korteweg-de Vries equation which has been investigated by many authors; for instance, see [5–11]. Kenig et al. [9] proved that the Cauchy problem for the modified KdV equation is locally well-posed in Hs(R) with s≥ 4 . Kenig et al. [10] proved that the Cauchy problem for the modified KdV equation is illposed in Hs(R) with s < 4 . Colliander et al. [6] proved that the Cauchy problem for the