Fractal solutions of dispersive partial differential equations on the torus
Analysis of PDEs
2018-09-21 v3 Number Theory
Abstract
We use exponential sums to study the fractal dimension of the graphs of solutions to linear dispersive PDE. Our techniques apply to Schr\"odinger, Airy, Boussinesq, the fractional Schr\"odinger, 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 Schr\"odinger and Korteweg-de Vries equations along oblique lines in space-time.
Cite
@article{arxiv.1803.00674,
title = {Fractal solutions of dispersive partial differential equations on the torus},
author = {Burak Erdoğan and George Shakan},
journal= {arXiv preprint arXiv:1803.00674},
year = {2018}
}
Comments
29 pages, v2: added references and discussion