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

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Abstract

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.

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Acknowledgements

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.

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Authors and Affiliations

  1. Department of Applied Mathematics and Statistics and Operations Research, Faculty of Science and Technology, University of the Basque Country UPV/EHU, Barrio Sarriena S/N, 48940, Leioa, Spain

    Francisco de la Hoz

  2. Department of Mathematics, Faculty of Science and Technology, University of the Basque Country UPV/EHU, Barrio Sarriena S/N, 48940, Leioa, Spain

    Luis Vega

  3. BCAM - Basque Center for Applied Mathematics, Alameda Mazarredo, 14, 48009, Bilbao, Spain

    Luis Vega

Authors
  1. Francisco de la Hoz
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  2. Luis Vega
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Correspondence to Luis Vega.

Additional information

Communicated by Alex Kiselev.

This work was supported by an ERCEA Advanced Grant 2014 669689 - HADE, by the MINECO Projects MTM2014-53850-P and SEV-2013-0323, and by the Basque Government Project IT641-13.

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de la Hoz, F., Vega, L. On the Relationship Between the One-Corner Problem and the M-Corner Problem for the Vortex Filament Equation. J Nonlinear Sci 28, 2275–2327 (2018). https://doi.org/10.1007/s00332-018-9477-7

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  • DOI: https://doi.org/10.1007/s00332-018-9477-7

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