Dispersive decay for the mass-critical generalized Korteweg-de Vries equation and generalized Zakharov-Kuznetsov equations

Abstract

In this paper, we discuss pointwise decay estimate for the solution to the mass-critical generalized Korteweg-de Vries (gKdV) equation with initial data $u_0\in H^{1/2}(\mathbb{R})$. It is showed that nonlinear solution enjoys the same decay rate as linear one. Moreover, we also quantify the decay for solutions to the generalized Zakharov-Kuznetsov equation which is a natural multi-dimensional extension of the gKdV equation. We obtain some decay estimates for nonlinear solutions to generalized Zakharov-Kuznetsov equations with small initial data in $H^2(\mathbb{R}^d)$.