On the Relationship Between the One-Corner Problem and the M-Corner Problem for the Vortex Filament Equation
Copyright policy )On the Relationship Between the One-Corner Problem and the M-Corner Problem for the Vortex Filament Equation
In this paper, we give evidence that the evolution of the Vortex Filament Equation for a regular $M$-corner polygon as initial datum can be explained at infinitesimal times as the superposition of $M$ one-corner initial data. Therefore, and due to periodicity, the evolution at later times can be understood as the nonlinear interaction of infinitely many filaments, one for each corner. 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.
33 pages, 13 figures
Transfer of Energy, vortex filament equation, NLS equations (nonlinear Schrödinger equations), turbulence, Talbot effect, Numerical Analysis (math.NA), Vortex flows for incompressible inviscid fluids, PDEs in connection with fluid mechanics, Talbot Effect, Second-order parabolic systems, Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations, Vortex methods applied to problems in fluid mechanics, Turbulence, Vortex filament equation, Fractals, multifractality, intermittency, FOS: Mathematics, Intermittency, Mathematics - Numerical Analysis, Numerical methods for discrete and fast Fourier transforms, transfer of energy, Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs, Multifractality
Transfer of Energy, vortex filament equation, NLS equations (nonlinear Schrödinger equations), turbulence, Talbot effect, Numerical Analysis (math.NA), Vortex flows for incompressible inviscid fluids, PDEs in connection with fluid mechanics, Talbot Effect, Second-order parabolic systems, Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations, Vortex methods applied to problems in fluid mechanics, Turbulence, Vortex filament equation, Fractals, multifractality, intermittency, FOS: Mathematics, Intermittency, Mathematics - Numerical Analysis, Numerical methods for discrete and fast Fourier transforms, transfer of energy, Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs, Multifractality
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