Dispersive Blow-up for Solutions of the Zakharov-Kuznetsov equation

TL;DR

This paper investigates dispersive blow-up phenomena in solutions of the Zakharov-Kuznetsov equation, demonstrating how singularities can form and persist in both linear and nonlinear cases.

Contribution

It constructs initial data leading to dispersive blow-up and shows that such singularities are inherited by solutions of the nonlinear Zakharov-Kuznetsov equation.

Findings

Linear solutions can develop point singularities due to focusing.

Nonlinear solutions inherit dispersive blow-up from linear components.

Results extend to generalized Zakharov-Kuznetsov equations.

Abstract

The main purpose here is the study of dispersive blow-up for solutions of the Zakharov-Kuznetsov equation. Dispersive blow-up refers to point singularities due to the focusing of short or long waves. We will construct initial data such that solutions of the linear problem present this kind of singularities. Then we show that the corresponding solutions of the nonlinear problem present dispersive blow-up inherited from the linear component part of the equation. Similar results are obtained for the generalized Zakharov-Kuznetsov equation.