2018
DOI: 10.1007/s00205-018-1300-y
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Relative Periodic Solutions of the N-Vortex Problem Via the Variational Method

Abstract: This article studies the N-vortex problem in the plane with positive vorticities. After an investigation of some properties for normalised relative equilibria of the system, we use symplectic capacity theory to show that, there exist infinitely many normalised relative periodic orbits on a dense subset of all energy levels, which are neither fixed points nor relative equilibria.

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Cited by 9 publications
(4 citation statements)
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“…For example in [35] periodic orbits for the n-vortex problem in the plane have been found. Unfortunately, it is a difficult problem to distinguish periodic orbits on a given energy level, when these are determined through implicit methods, instead of explicit constructions.…”
Section: Absolute and Relative Periodic Orbitsmentioning
confidence: 99%
See 1 more Smart Citation
“…For example in [35] periodic orbits for the n-vortex problem in the plane have been found. Unfortunately, it is a difficult problem to distinguish periodic orbits on a given energy level, when these are determined through implicit methods, instead of explicit constructions.…”
Section: Absolute and Relative Periodic Orbitsmentioning
confidence: 99%
“…The analogues in vortex problems without the harmonic trap are studied by [27,31]. The following lemma is proved by using the conservation of centre of vorticity as is in [35].…”
Section: B1 Shub's Lemma For N-vortex In Bose Einstein Condensationmentioning
confidence: 99%
“…Numerous strategies and their applications are accessible in the literature to compute the exact solutions of different classes of DEs that arise in numerous fields of science. There is no proficient and widespread methodology so far that a wide range of nonlinear equations can computed, for example the variational technique [8], ¢ G G-expansion technique [9,10], new extended direct algebraic technique [11], the asymptotic technique [12], the inverse scattering technique [13], the first integral method [14], soliton perturbation theory [15], and some other methods are represented in [16][17][18][19][20][21][22][23][24][25].…”
Section: Introductionmentioning
confidence: 99%
“…A role diametrically opposed to singular configurations is played by relative equilibrium configurations, in which point vortices evolve while their relative positions remain fixed. This is in fact a very active area of research: we limit ourselves to refer to some recent works [29,31,13,12,36] and references therein. For the sake of completeness, let us also recall that configurations leading to quasi-periodic vortex dynamics have been proved to form a subset of positive measure of phase space in [20], by means of KAM theory.…”
Section: Introductionmentioning
confidence: 99%