Vortex filament equation for a regular polygon. (English) Zbl 1339.35300

de la Hoz, Francisco; Vega, Luis

Nonlinearity 27, No. 12, 3031-3057 (2014).

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.

Reviewer: Yvan Martel (Palaiseau)

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

Keywords:

vortex filament equation; Schrödinger map; quadratic Gauss sums; pseudo spectral methods; fractals