Fractal solutions of dispersive partial differential equations on the torus

Abstract

We use exponential sums to study the fractal dimension of the graphs of solutions to linear dispersive PDE. Our techniques apply to Schrodinger, Airy, Boussinesq, the fractional Schrodinger, 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 Schrodinger and Korteweg–de Vries equations along oblique lines in space–time.