Dispersive blow-up for solutions to three dimensional generalized Zakharov-Kuznetsov equations

We illustrate the dispersive blow up phenomena of the solutions of three dimensional generalized Zakharov-Kuznetsov equations. In particular, we construct smooth initial data such that, the associated global solutions fail to be $C^1$ at time $t$ in a null set containing all rational numbers, but are $C^1$ at all times $t$ which are generic irrational numbers. The key ingredient are to construct linear solutions which exhibit such phenomena and to prove nonlinear smoothing estimates for the full nonlinear model.