Fractal solutions of dispersive partial differential equations on the torus

2018/03/01 by Burak Erdoğan, Erdoğan, Burak, George Shakan +1
Mathematics · Physics and Astronomy · #11L07 #35A99 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Black Holes and Theoretical Physics #FOS: Mathematics #Nonlinear Waves and Solitons #Number Theory (math.NT) #Orbital Angular Momentum in Optics #math.AP #math.NT #msc:11L07 #msc:35A99

paper · pdf · doi:10.48550/arxiv.1803.00674

29 pages, v2: added references and discussion

openalex publication_date 2018/03/01 · arxiv created 2018/09/20 · arxiv updated 2018/09/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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

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