Well-posedness and small data scattering for the generalized Ostrovsky equation

We consider the generalized Ostrovsky equation utx=u+(up)xx. We show that the equation is locally well posed in Hs, s>3/2 for all integer values of p⩾2. For p⩾4, we show that the equation is globally well posed for small data in H5∩W3,1 and moreover, it scatters small data. The latter results are corroborated by numerical computations which confirm the heuristically expected decay of ‖u‖Lr∼t−(r−2)/(2r).