LAUSR

We consider the Cauchy problem for the generalized Zakharov–Kuznetzov equation $$\partial _t u + \partial _x \Delta u = \partial _x ( u^{m+1} )$$∂tu+∂xΔu=∂x(um+1) on two or three space dimensions. We mainly study the two dimensional case and give the local well-posedness and the small data global well-posedness in the modulation space $$M_{2,1}(\mathbb {R}^2)$$M2,1(R2) for $$m \ge 4$$m≥4. Moreover, for the quartic case (namely, $$m = 3$$m=3), the local well-posedness in $$ M_{2,1}^{1/4}(\mathbb {R}^2)$$M2,11/4(R2) is given. The well-posedness on three dimensions is also considered.