A note on the 2D generalized Zakharov–Kuznetsov equation: Local, global, and scattering results

We consider the generalized two-dimensional Zakharov–Kuznetsov equation ut+∂xΔu+∂x(uk+1)=0, where k⩾3 is an integer number. For k⩾8 we prove local well-posedness in the L2-based Sobolev spaces Hs(R2), where s is greater than the critical scaling index sk=1−2/k. For k⩾3 we also establish a sharp criteria to obtain global H1(R2) solutions. A nonlinear scattering result in H1(R2) is also established assuming the initial data is small and belongs to a suitable Lebesgue space.