On dispersive quantization and fractalization for the Kawahara equation

TL;DR

This paper explores the unique dispersive and fractal behaviors of solutions to the Kawahara equation, revealing a dichotomy where solutions are quantized at rational times and fractal at irrational times, supported by smoothing estimates.

Contribution

It provides the first analysis of the Talbot effect for the Kawahara equation, combining smoothing estimates with known linear results to describe solution behaviors.

Findings

Solutions are quantized at rational times.

Solutions exhibit fractal profiles at irrational times.

Mathematical description of the Talbot effect for the Kawahara equation.

Abstract

In this paper, we investigate the dichotomous behavior of solutions to the Kawahara equation with bounded variation initial data, analogous to the Talbot effect. Specifically, we observe that the solution is quantized at rational times, whereas at irrational times, it is a nowhere continuous differentiable function with a fractal profile. This phenomenon, however, has not been explored for the Kawahara equation, which is a fifth-order KdV type equation. To achieve this, we derive smoothing estimates for the nonlinear Duhamel solution, which, when combined with the known results on the linear solution, provides a mathematical description of the Talbot effect.