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Vortex filament equation for a regular polygon. (English) Zbl 1339.35300
The authors study both theoretically and numerically the evolution of the vortex filament equation (or binormal equation) \(X_t = X_s\wedge X_{ss}\) with a regular planar polygon \(X(s,0)\) as initial configuration, i.e. in the case of filaments with several corners. The authors complete and reinforce observations by Jerrard and Smets on the time-recurrent property of polygon configurations. They also related the behavior of \(X(0,t)\) to the Riemann’s non differential function, whose multi-fractal nature was proved by Jaffard.

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
11L05 Gauss and Kloosterman sums; generalizations
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
15A63 Quadratic and bilinear forms, inner products
28A80 Fractals
51E12 Generalized quadrangles and generalized polygons in finite geometry
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