Fractal solutions of dispersive partial differential equations on the torus

TL;DR

This paper investigates the fractal dimensions of solutions to various linear and nonlinear dispersive PDEs on the torus using exponential sums, providing bounds and insights into their geometric complexity.

Contribution

It introduces a novel application of exponential sums to analyze the fractal dimension of dispersive PDE solutions, extending to nonlinear cases and multiple equations.

Findings

Bounds for the fractal dimension of solutions to Schrödinger and KdV equations.

Application of techniques to a range of linear dispersive PDEs.

Insights into the geometric complexity of solutions along oblique space-time lines.

Abstract

We use exponential sums to study the fractal dimension of the graphs of solutions to linear dispersive PDE. Our techniques apply to Schr\"odinger, Airy, Boussinesq, the fractional Schr\"odinger, and the gravity and gravity-capillary water wave equations. We also discuss applications to certain nonlinear dispersive equations. In particular, we obtain bounds for the dimension of the graph of the solution to cubic nonlinear Schr\"odinger and Korteweg-de Vries equations along oblique lines in space-time.