Fractal solutions of linear and nonlinear dispersive partial differential equations

In this paper, we study fractal solutions of linear and nonlinear dispersive partial differential equation on the torus. In the first part, we answer some open questions on the fractal solutions of linear Schrodinger equation and equations with higher-order dispersion. We also discuss applications to their nonlinear counterparts like the cubic Schrodinger equation and the Korteweg-de Vries equation. In the second part, we study fractal solutions of the vortex filament equation and the associated Schrodinger map equation. In particular, we construct global strong solutions of the SM in H-s for s > 3/2 for which the evolution of the curvature is given by a periodic nonlinear Schrodinger evolution. We also construct unique weak solutions in the energy space H-1. Our analysis follows the frame construction of Chang, Shatah and Uhlenbeck ['Schrodinger maps', Comm. Pure Appl. Math. 53 (2000) 590-602] and Shatah, Vega and Zeng ['Schrodinger maps and their associated frame systems', Int. Math. Res. Not. (2007)].