Fractal solutions of linear and nonlinear dispersive partial differential equations

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 Schrödinger equation and equations with higher order dispersion. We also discuss applications to their nonlinear counterparts like the cubic Schrödinger 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 Schrödinger map equation (SM). In particular, we construct global strong solutions of the SM in $H^s$ for $s>\frac{3}{2}$ for which the evolution of the curvature is given by a periodic nonlinear Schrödinger 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}.

28 pages

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• New Revival Phenomena for Bidirectional Dispersive Hyperbolic Equations
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• Multifractality and intermittency in the limit evolution of polygonal vortex filaments