Relative Periodic Solutions of the N-Vortex Problem Via the Variational Method

Wang, Qun

doi:10.1007/s00205-018-1300-y

Cited by 9 publications

(4 citation statements)

References 40 publications

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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%

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Choreographic Holomorphic Spheres with Application to Hamiltonian Systems of $N$-Vortex Type

Wang

2018

Preprint

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We study the existence of (relative) simple choreographies for a class of Hamiltonian systems describing the interaction of particles in the plane motivated mainly by the nvortex type problem. In particular, by constructing choreographic pseudo-holomorphic spheres, we prove that there exist infinitely many non-trivial relative choreographies for the identical n-vortex problem arising from both the Euler equation and the Gross-Pitaevskii equation.

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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%

Choreographic Holomorphic Spheres with Application to Hamiltonian Systems of $N$-Vortex Type

Wang

2018

Preprint

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“…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%

Conserved quantities and travelling wave profiles to the nonlinear transmission line via Lie group analysis

Riaz¹,

Jhangeer²,

Abualnaja³

et al. 2021

Phys. Scr.

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The nonlinear transmission line (NLTL) equations are significant nonlinear evolution equations (NLEEs) in nonlinear electrical transmission line (NLETL) regulation. The tanh method is employed to compute some traveling wave patterns of the NLTL equation. The new extended direct algebraic method (NEDAM) is a viable and successful mathematical method to construct the traveling wave patterns of science and engineering problems. The NEDAM is effectively utilized to obtain the traveling wave structures of a considered model in the form of trigonometric and hyperbolic functions containing parameters. The Lie symmetry technique is used to analyze the NLTL equation and compute the Infinitesimal generators. Moreover, we have shown graphically obtained wave profiles by using the different suitable values of the parameters involved. Further, the nonlinear transmission line equation is described through nonlinear self-adjointness, and conserved quantities are computed for each vector.

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“…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%

Burst of Point Vortices and Non-Uniqueness of 2D Euler Equations

Grotto,

Pappalettera

2020

Preprint

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We give a rigorous construction of solutions to the Euler point vortices system in which three vortices burst out of a single one in a configuration of many vortices, or equivalently that there exist configurations of arbitrarily many vortices in which three of them collapse in finite time. As an intermediate step, we show that well-known self-similar bursts and collapses of three isolated vortices in the plane persist under a sufficiently regular external perturbation. We also discuss how our results produce examples of non-unique weak solutions to 2-dimensional Euler's equations -in the sense introduced by Schochet-in which energy is dissipated.

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Relative Periodic Solutions of the N-Vortex Problem Via the Variational Method

Cited by 9 publications

References 40 publications

Choreographic Holomorphic Spheres with Application to Hamiltonian Systems of $N$-Vortex Type

Choreographic Holomorphic Spheres with Application to Hamiltonian Systems of $N$-Vortex Type

Conserved quantities and travelling wave profiles to the nonlinear transmission line via Lie group analysis

Burst of Point Vortices and Non-Uniqueness of 2D Euler Equations

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