On the Cauchy problems associated to a ZK-KP-type family equations with a transversal fractional dispersion

In this paper we examine the well-posedness and ill-posedeness of the Cauchy problems associated to a family equations of ZK-KP-type $$\{\begin{array}{l}{u}_t=u_{xxx}-\mathcal{H}{D}_x^{\alpha }{u}_{yy}+u{u}_x,\\ u(0)=\psi \in Z\end{array}$$ in anisotropic Sobolev spaces, where $1\le \alpha \le 1$, $\mathcal{H}$ is the Hilbert transform and $D_x^{\alpha }$ is the fractional derivative, both with respect to $x$.