Regular Polygonal Vortex Filament Evolution and Exponential Sums
When taking a regular planar polygon of M sides and length 2π as the initial datum of the vortex filament equation, Xt = Xs ∧Xss , the solution becomes polygonal at times of the form tpq = (p/q)(2π/M2), with gcd(p, q) = 1, and the corresponding polygon has Mq sides, if q is odd, and Mq/2 sides, if q...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2024 |
| País: | España |
| Institución: | Universidad del País Vasco |
| Repositorio: | Addi. Archivo Digital para la Docencia y la Investigación |
| OAI Identifier: | oai:addi.ehu.eus:10810/70804 |
| Acceso en línea: | http://hdl.handle.net/10810/70804 |
| Access Level: | acceso abierto |
| Palabra clave: | vortex filament equation nonlinear Schrödinger equation rotation matrices trigonometric sums |
| Sumario: | When taking a regular planar polygon of M sides and length 2π as the initial datum of the vortex filament equation, Xt = Xs ∧Xss , the solution becomes polygonal at times of the form tpq = (p/q)(2π/M2), with gcd(p, q) = 1, and the corresponding polygon has Mq sides, if q is odd, and Mq/2 sides, if q is even. Moreover, that polygon is skew (except when q = 1 or q = 2, where the initial shape is recovered), and the angle ρ between two adjacent sides is a constant. In this paper, we give a rigorous proof of the conjecture that states that, at a time tpq , cosq (ρ/2) = cos(π/M), if q is odd, and cosq (ρ/2) = cos 2(π/M), if q is even. Since the transition of one side of the polygon to the next one is given by a rotation in R3 determined by a generalized Gauss sum, the idea of the proof consists in showing that a certain product of those rotations is a rotation of angle 2π/M, which is equivalent to proving that some exponential sums with arithmetic content are purely imaginary. |
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