Fractal solutions of linear and nonlinear dispersive partial differential equations

TL;DR

This paper investigates fractal solutions of linear and nonlinear dispersive PDEs on the torus, answering open questions, constructing solutions, and exploring applications to equations like NLS, KdV, and vortex filament equations.

Contribution

It provides new insights into fractal solutions of dispersive PDEs, including solutions for linear, nonlinear, and geometric equations, with constructions and applications.

Findings

Solved open questions on linear Schrödinger fractal solutions

Constructed global strong solutions for Schrödinger map equations

Analyzed fractal solutions in nonlinear dispersive PDEs

Abstract

In this paper we study fractal solutions of linear and nonlinear dispersive PDE on the torus. In the first part we answer some open questions on the fractal solutions of linear Schr\"odinger equation and equations with higher order dispersion. We also discuss applications to their nonlinear counterparts like the cubic Schr\"odinger equation (NLS) and the Korteweg-de Vries equation (KdV). In the second part, we study fractal solutions of the vortex filament equation and the associated Schr\"odinger map equation (SM). In particular, we construct global strong solutions of the SM in $H^s$ for $s>\frac32$ for which the evolution of the curvature is given by a periodic nonlinear Schr\"odinger evolution. We also construct unique weak solutions in the energy level. Our analysis follows the frame construction of Chang {\em et al.} \cite{csu} and Nahmod {\em et al.} \cite{nsvz}.