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, 25:11 | Cite as

Fractal solutions of dispersive partial differential equations on the torus

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

Mathematics Subject Classification

35Q55 11L03 

Notes

References

  1. 1.
    Berry, M.V.: Quantum fractals in boxes. J. Phys. A Math. Gen. 29, 6617–6629 (1996)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Berry, M.V., Klein, S.: Integer, fractional and fractal Talbot effects. J. Mod. Opt. 43, 2139–2164 (1996)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Berry, M.V., Lewis, Z.V.: On the Weierstrass–Mandelbrot fractal function. Proc. R. Soc. Lond. A 370, 459–484 (1980)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Berry, M.V., Marzoli, I., Schleich, W.: Quantum carpets, carpets of light. Phys. World 14(6), 39–44 (2001)CrossRefGoogle Scholar
  5. 5.
    Bourgain, J.: Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Geom. Funct. Anal. 3, 107–156 (1993)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bourgain, J.: Decoupling, exponential sums and the Riemann zeta function. J. Am. Math. Soc. 30(1), 205–224 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bourgain, J., Demeter, C., Guth, L.: Proof of the main conjecture in Vinogradov’s mean value theorem for degrees higher than three. Ann. Math. (2) 184(2), 633–682 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chamizo, F., Cordoba, A.: Differentiability and dimension of some fractal Fourier series. Adv. Math. 142, 335–354 (1999)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chen, G., Olver, P.J.: Dispersion of discontinuous periodic waves. Proc. R. Soc. Lond. A 469, 20120407 (2012)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Chen, G., Olver, P.J.: Numerical simulation of nonlinear dispersive quantization. Discrete Contin. Dyn. Syst. 34(3), 991–1008 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Chousionis, V., Erdoğan, M.B., Tzirakis, N.: Fractal solutions of linear and nonlinear dispersive partial differential equations. Proc. Lond. Math. Soc. (3) 110, 543–564 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    de la Hoz, F., Vega, L.: Vortex filament equation for a regular polygon. Nonlinearity 27(12), 3031–3057 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Deliu, A., Jawerth, B.: Geometrical dimension versus smoothness. Constr. Approx. 8, 211–222 (1992)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Demirbaş, S., Erdoğan, M.B., Tzirakis, N.: Existence and uniqueness theory for the fractional Schrödinger equation on the torus, some topics in harmonic analysis and applications. Advanced Lectures in Mathematics (ALM), vol. 34, pp. 145–162. International Press, Somerville (2016)Google Scholar
  15. 15.
    Erdoğan, M.B., Gürel, B., Tzirakis, N.: Smoothing for the fractional Schrödinger equation on the torus and the real line. Accepted by Indiana Univ. Math. JGoogle Scholar
  16. 16.
    Erdoğan, M.B., Tzirakis, N.: Global smoothing for the periodic KdV evolution. Int. Math. Res. Not. (2012). https://doi.org/10.1093/imrn/rns189
  17. 17.
    Erdoğan, M.B., Tzirakis, N.: Talbot effect for the cubic nonlinear Schrödinger equation on the torus. Math. Res. Lett. 20(6), 1081–1090 (2013)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Erdoğan, M.B., Tzirakis, N.: Dispersive Partial Differential Equations: Wellposedness and Applications, London Mathematical Society Student Texts, vol. 86. Cambridge University Press, Cambridge (2016)CrossRefGoogle Scholar
  19. 19.
    Graham, S.W., Kolesnik, G.: Van der Corput’s Method of Exponential Sums, London Mathematical Society Lecture Note Series, vol. 126. Cambridge University Press, Cambridge (1991)CrossRefGoogle Scholar
  20. 20.
    Heath-Brown, D.R.: A new k-th derivative estimate for exponential sums via Vinogradov’s mean value, Tr. Mat. Inst. Steklova, Analiticheskaya i Kombinatornaya Teoriya Chisel, vol. 296, pp. 95–110 (2017)Google Scholar
  21. 21.
    Iwaniec, H., Kowalski, E.: Analytic Number Theory. AMS Colloquium Publications, AMS, Providence (2004)zbMATHGoogle Scholar
  22. 22.
    Jaffard, S.: The spectrum of singularities of Riemann’s function. Revista Matemátic Iberoamericana 12, 441–460 (1996)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Kapitanski, L., Rodnianski, I.: Does a quantum particle knows the time? In: Hejhal, D., Friedman, J., Gutzwiller, M.C., Odlyzko, A.M. (eds.) Emerging Applications of Number Theory, IMA Volumes in Mathematics and Its Applications, vol. 109, pp. 355–371. Springer, New York (1999)CrossRefGoogle Scholar
  24. 24.
    Katznelson, Y.: An Introduction to Harmonic Analysis. Cambridge Mathematical Library, 3rd edn. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  25. 25.
    Khinchin, A.Y.: Continued fractions. The 3rd Russian edition of 1961 (in Translation). The University of Chicago Press, Chicago (1964)Google Scholar
  26. 26.
    Lévy, P.: Théorie de l’addition des variables aléatoires. Gauthier-Villars, Paris (1937)zbMATHGoogle Scholar
  27. 27.
    Olver, P.J.: Dispersive quantization. Am. Math. Mon. 117(7), 599–610 (2010)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Olver, P.J., Sheils, N.E.: Dispersive Lamb Systems. Preprint 2017. arXiv:1710.05814v1
  29. 29.
    Olver, P.J., Tsatis, E.: Points of constancy of the periodic linearized Korteweg–deVries equation. Preprint 2018. arXiv:1802.01213v1
  30. 30.
    Oskolkov, K.I.: A class of I. M. Vinogradov’s series and its applications in harmonic analysis. In: Gonchar, A.A., Saff, E.B. (eds.) Progress in Approximation Theory (Tampa, FL, 1990), Springer Series in Computational Mathematics, vol. 19, pp. 353–402. Springer, New York (1992)Google Scholar
  31. 31.
    Oskolkov, K.I.: The Schrödinger Density and the Talbot Effect. Approximation and Probability, Banach Center Publications, 72, pp. 189–219. Institute of Mathematics of Polish Academy of Science, Warsaw (2006)Google Scholar
  32. 32.
    Oskolkov, K.I., Chakhkiev, M.A.: On the “nondifferentiable” Riemann function and the Schrödinger equation. (Russian) Tr. Mat. Inst. Steklova, Teoriya Funktsii i Differentsialnye Uravneniya, 269, pp. 193–203 (2010); Proc. Steklov Inst. Math. 269, no. 1, pp. 186–196 (2010) (in translation)Google Scholar
  33. 33.
    Oskolkov, K.I., Chakhkiev, M.A.: Traces of the discrete Hilbert transform with quadratic phase. (Russian) Tr. Mat. Inst. Steklova, Ortogonalnye Ryady, Teoriya Priblizheni i Smezhnye Voprosy, vol. 280, pp. 255–269 (2013); Proc. Steklov Inst. Math. 280, no. 1, pp. 248–262 (2013) (in translation)Google Scholar
  34. 34.
    Rodnianski, I.: Fractal solutions of the Schrödinger equation. Contemp. Math. 255, 181–187 (2000)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Stein, E.M., Shakarchi, R.: Real Analysis. Measure Theory, Integration, and Hilbert Spaces. Princeton University Press, Princeton (2005)zbMATHGoogle Scholar
  36. 36.
    Talbot, H.F.: Facts related to optical science. Philos. Mag. 9, 401–407 (1836)Google Scholar
  37. 37.
    Taylor, M.: Tidbits in Harmonic Analysis. Lecture Notes. UNC, Chapel Hill (1998)Google Scholar
  38. 38.
    Taylor, M.: The Schrödinger equation on spheres. Pac. J. Math. 209, 145–155 (2003)CrossRefGoogle Scholar
  39. 39.
    Triebel, H.: Theory of Function Spaces. Birkhäuser, Basel (1983)CrossRefGoogle Scholar
  40. 40.
    Vaughan, R.C.: The Hardy–Littlewood Method, Cambridge Tracts in Mathematics, vol. 80. Cambridge University Press, Cambridge-New York. ISBN: 0-521-23439-5 (1981)Google Scholar
  41. 41.
    Vega, L.: The dynamics of vortex filaments with corners. Commun. Pure Appl. Anal. 14(4), 1581–1601 (2015)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Wooley, T.D.: Nested efficient congruencing and relatives of Vinogradovs mean value theorem. PreprintGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of IllinoisUrbanaUSA

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