Fractal solutions of dispersive partial differential equations on the torus

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...

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Bibliographic Details
Published inSelecta mathematica (Basel, Switzerland) Vol. 25; no. 1; pp. 1 - 26
Main Authors Erdoğan, M. B., Shakan, G.
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.03.2019
Springer Nature B.V
Subjects
Boussinesq equations
Capillary waves
Fractals
Gravitation
Mathematical analysis
Mathematics
Mathematics and Statistics
Nonlinear equations
Partial differential equations
Toruses
Water waves
Wave equations
35Q55
11L03
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Summary: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.
ISSN:1022-1824
1420-9020
DOI:10.1007/s00029-019-0455-1