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Fractal solutions of dispersive partial differential equations on the torus

  • M. B. ErdoğanEmail author
  • G. ShakanEmail author


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.

Mathematics Subject Classification

35Q55 11L03 



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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of IllinoisUrbanaUSA

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