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Fractal solutions of dispersive partial differential equations on the torus

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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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  1. Department of Mathematics, University of Illinois, Urbana, IL, 61801, USA

    M. B. Erdoğan & G. Shakan

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  1. M. B. Erdoğan
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  2. G. Shakan
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M. B. Erdoğan is partially supported by NSF Grant DMS-1501041. G. Shakan was partially supported by NSF Grant DMS-1501982 and would like to thank Kevin Ford for financial support. The authors would like to thank Luis Vega for useful comments on an earlier version of this manuscript and for pointing out several important references. The authors also thank an anonymous referee for useful suggestions and references.

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Erdoğan, M.B., Shakan, G. Fractal solutions of dispersive partial differential equations on the torus. Sel. Math. New Ser. 25, 11 (2019). https://doi.org/10.1007/s00029-019-0455-1

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