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Vortex Filament Equation for a Regular Polygon in the Hyperbolic Plane.

Francisco de la Hoz1, Sandeep Kumar2,3, Luis Vega1,2

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

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Summary

This study analyzes the vortex filament equation for polygons in hyperbolic space, finding numerical and algebraic solutions align. It also shows polygon evolution relates to single-corner data, enabling comparisons with Euclidean cases.

Keywords:
Hyperbolic planeMultifractalitySchrödinger mapTalbot effectVortex filament equation

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Area of Science:

  • Differential Geometry
  • Mathematical Physics
  • Computational Mathematics

Background:

  • The vortex filament equation (VFE) describes the motion of curves in space.
  • Studying VFE in hyperbolic space presents unique challenges due to open, exponentially growing polygon ends.
  • Previous research often focused on Euclidean space or simpler geometric shapes.

Purpose of the Study:

  • To analyze the evolution of a regular planar polygon under the VFE in hyperbolic space.
  • To validate numerical methods against algebraic techniques for VFE in this non-Euclidean setting.
  • To establish a relationship between the evolution of a planar polygon and superposition of one-corner initial data.

Main Methods:

  • A finite difference scheme in space was employed.
  • A fourth-order Runge-Kutta method was used for time integration.
  • Fixed boundary conditions were applied, and results were compared with algebraic solutions.

Main Results:

  • Numerical solutions for the VFE in hyperbolic space were found to be in complete agreement with algebraic techniques.
  • The evolution of a planar polygon at infinitesimal times was shown to be a superposition of one-corner initial data.
  • The speed of the center of mass of the polygon was computed.

Conclusions:

  • The study successfully demonstrates a numerical approach for VFE in hyperbolic space, validating its accuracy.
  • The findings provide a new perspective on polygon dynamics by linking them to simpler initial data configurations.
  • This work facilitates comparisons of geometric evolution between hyperbolic and Euclidean spaces.