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 Schrödinger equation and equations with higher-order dispersion. We also discuss applications to their nonlinear counterparts like the cubic Schrödinger equation and the Korteweg-de Vries equation. In the second part, we study fractal solutions of the vortex filament equation and the associated Schrödinger map equation. In particular, we construct global strong solutions of the SM in Hs for s > 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 space H1. Our analysis follows the frame construction of Chang, Shatah and Uhlenbeck ['Schrödinger maps', Comm. Pure Appl. Math. 53 (2000) 590-602] and Shatah, Vega and Zeng ['Schrödinger maps and their associated frame systems', Int. Math. Res. Not. (2007)].
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