Journal of Nonlinear Science

, Volume 28, Issue 6, pp 2275–2327 | Cite as

On the Relationship Between the One-Corner Problem and the M-Corner Problem for the Vortex Filament Equation

  • Francisco de la Hoz
  • Luis VegaEmail author


In this paper, we give evidence that the evolution of the vortex filament equation (VFE) for a regular M-corner polygon as initial datum can be explained at infinitesimal times as the superposition of M one-corner initial data. This fact is mainly sustained with the calculation of the speed of the center of mass; in particular, we show that several conjectures made at the numerical level are in agreement with the theoretical expectations. Moreover, due to the spatial periodicity, the evolution of VFE at later times can be understood as the nonlinear interaction of infinitely many filaments, one for each corner; and this interaction turns out to be some kind of nonlinear Talbot effect. We also give very strong numerical evidence of the transfer of energy and linear momentum for the M-corner case; and the numerical experiments carried out provide new arguments that support the multifractal character of the trajectory defined by one of the corners of the initial polygon.


Intermittency Multifractality Talbot Effect Transfer of Energy Turbulence Vortex filament equation 

Mathematics Subject Classification

11L05 28A80 35Q55 65M70 65T50 76B47 



We want to thank V. Banica and C. García-Cervera for very enlightening conversations concerning the last two sections of this paper. Part of this work was started while the second author was visiting MSRI, within the New Challenges in PDE 2015 program. We also want to thank the anonymous reviewers for their very valuable comments.


  1. Arms, R.J., Hama, F.R.: Localized-induction concept on a curved vortex and motion of an elliptic vortex ring. Phys. Fluids 8(4), 553–559 (1965)CrossRefGoogle Scholar
  2. Banica, V., Vega, L.: On the stability of a singular vortex dynamics. Commun. Math. Phys. 286(2), 593–627 (2009)MathSciNetCrossRefGoogle Scholar
  3. Banica, V., Vega, L.: Scattering for 1D cubic NLS and singular vortex dynamics. J. Eur. Math. Soc. (JEMS) 14(1), 209–253 (2012)MathSciNetCrossRefGoogle Scholar
  4. Banica, V., Vega, L.: Stability of the Self-similar dynamics of a vortex filament. Arch. Ration. Mech. Anal. 210(3), 673–712 (2013)MathSciNetCrossRefGoogle Scholar
  5. Banica, V., Vega, L.: The initial value problem for the Binormal Flow with rough data. Ann. Sci. l’ENS 48(6), 1423–1455 (2015)MathSciNetzbMATHGoogle Scholar
  6. Banica, V., Vega, L.: Singularity formation for the 1-D cubic NLS and the Schrödinger map on \(\mathbb{S}^2\). Commun. Pure Appl. Anal. 17(4), 1317–1329 (2018)MathSciNetCrossRefGoogle Scholar
  7. Berry, M.V., Klein, S.: Integer, fractional and fractal Talbot effects. J. Mod. Opt. 43, 2139–2164 (1996)MathSciNetCrossRefGoogle Scholar
  8. Buttke, T.F.: A numerical study of superfluid turbulence in the self-induction approximation. J. Comput. Phys. 76(2), 301–326 (1998)MathSciNetCrossRefGoogle Scholar
  9. Chen, G., Olver, P.J.: Dispersion of discontinuous periodic waves. Proc. R. Soc. Lond. A 469, 20120407 (2012)MathSciNetCrossRefGoogle Scholar
  10. Chen, G., Olver, P.J.: Numerical simulation of nonlinear dispersive quantization. Discrete Contin. Dyn. Syst. 34(3), 991–1008 (2014)MathSciNetCrossRefGoogle Scholar
  11. Chousionis, V., Erdoğan, M.B., Tzirakis, N.: Fractal solutions of linear and nonlinear dispersive partial differential equations. Proc. Lond. Math. Soc. 110(3), 543–564 (2015)MathSciNetCrossRefGoogle Scholar
  12. Da Rios, L.S.: Sul moto d’un liquido indefinito con un filetto vorticoso di forma qualunque. Rend. Circ. Mat. Palermo 22(1), 117–135 (1906). (In Italian) CrossRefGoogle Scholar
  13. de la Hoz, F.: Self-similar solutions for the 1-D Schrödinger map on the hyperbolic plane. Math. Z. 257(1), 61–80 (2007)MathSciNetCrossRefGoogle Scholar
  14. de la Hoz, F., García-Cervera, C.J., Vega, L.: A numerical study of the self-similar solutions of the Schrödinger map. SIAM J. Appl. Math. 70(4), 1047–1077 (2009)MathSciNetCrossRefGoogle Scholar
  15. de la Hoz, F., Vega, L.: Vortex filament equation for a regular polygon. Nonlinearity 27(12), 3031–3057 (2014)MathSciNetCrossRefGoogle Scholar
  16. de la Hoz, F., Vega, L.: The vortex filament equation as a pseudorandom generator. Acta Appl. Math. 138(1), 135–151 (2015)MathSciNetCrossRefGoogle Scholar
  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. Fonda, E., Meichle, D.P., Ouellette, N.T., Hormoz, S., Lathrop, D.P.: Direct observation of Kelvin waves excited by quantized vortex reconnection. PNAS 111(Suppl. 1), 4707–4710 (2014)CrossRefGoogle Scholar
  19. Grinstein, F.F., Gutmark, E.J.: Flow control with noncircular jets. Ann. Rev. Fluid Mech. 31, 239–272 (1999)CrossRefGoogle Scholar
  20. Grinstein, F.F., Gutmark, E.J., Parr, T.: Near field dynamics of subsonic, free square jets. A computational and experimental study. Phys. Fluids 7, 1483–1497 (1995)CrossRefGoogle Scholar
  21. Gutiérrez, S., Rivas, J., Vega, L.: Formation of singularities and self-similar vortex motion under the localized induction approximation. Commun. PDE 28(5–6), 927–968 (2003)MathSciNetCrossRefGoogle Scholar
  22. Hasimoto, H.: A soliton on a vortex filament. J. Fluid Mech. 51(3), 477–485 (1972)MathSciNetCrossRefGoogle Scholar
  23. Ishimori, Y.: An integrable classical spin chain. J. Phys. Soc. Jpn. 51(11), 3417–3418 (1982)MathSciNetCrossRefGoogle Scholar
  24. Jaffard, S.: The spectrum of singularities of Riemann’s function. Rev. Mat. Iberoam. 12(2), 441–460 (1996)MathSciNetCrossRefGoogle Scholar
  25. Jerrard, R.L., Seis, C.: On the vortex filament conjecture for Euler flows. Arch. Ration. Mech. Anal. 224(1), 135–172 (2017)MathSciNetCrossRefGoogle Scholar
  26. Jerrard, R.L., Smets, D.: On Schrödinger maps from \(T^1\) to \(S^2\). Ann. Sci. Éc. Norm. Supér. (4) 45(4), 637–680 (2013)CrossRefGoogle Scholar
  27. Jerrard, R.L., Smets, D.: On the motion of a curve by its binormal curvature. J. Eur. Math. Soc. 17(6), 1487–1515 (2015)MathSciNetCrossRefGoogle Scholar
  28. Lakshmanan, M.: The fascinating world of the Landau–Lifshitz–Gilbert equation: an overview. Philos. Trans. R. Soc. A 369, 1280–1300 (2011)MathSciNetCrossRefGoogle Scholar
  29. Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flows, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002)Google Scholar
  30. Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W.: NIST Handbook of Mathematical Functions. NIST and Cambridge University Press, Cambridge (2010)zbMATHGoogle Scholar
  31. Olver, P.J.: Dispersive quantization. Am. Math. Mon. 117(7), 599–610 (2010)MathSciNetCrossRefGoogle Scholar
  32. Ricca, R.L.: Physical interpretation of certain invariants for vortex filament motion under LIA. Phys. Fluids A 4, 938–944 (1992)MathSciNetCrossRefGoogle Scholar
  33. Saffman, P.G.: Vortex Dynamics, Cambridge Monographs on Mechanics. Cambridge University Press, Cambridge (1995)Google Scholar
  34. Zhang, Y., Wen, J., Zhu, S.N., Xiao, M.: Nonlinear Talbot effect. Phys. Rev. Lett. 104(18), 183901 (2010)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Applied Mathematics and Statistics and Operations Research, Faculty of Science and TechnologyUniversity of the Basque Country UPV/EHULeioaSpain
  2. 2.Department of Mathematics, Faculty of Science and TechnologyUniversity of the Basque Country UPV/EHULeioaSpain
  3. 3.BCAM - Basque Center for Applied MathematicsBilbaoSpain

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