Title: Symmetry analysis of equilibrium solutions for a pair of interacting vortex filaments

Authors: Mohamed I. Jamaloodeen

Addresses: School of Science and Technology, Georgia Gwinnett College, 1000 University Center Lane, Lawrenceville, GA 30043, USA

Abstract: Using symmetry methods, an important technique in nonlinear analysis, we analyse equilibrium solutions for a pair of interacting vortex filaments. We characterise all fixed equilibrium (stationary) solutions. Through symmetry, we then identify classes of relative equilibrium solutions, in which the distances between the two vortices are time-independent and compare with known plane wave relative equilibria. Using Lie symmetry group analysis, we show how known fixed equilibria, when Γ = Γ₁ = Γ₂ = 1, give rise to relative equilibria. We show, when Γ = Γ₁ = −Γ₂ = 1, that known relative equilibria generate additional relative equilibria through linear displacements in the spatial variable. We finally consider more symmetrical configurations, in which the vortex filaments interact self-similarly. We find a rich class of self-similar equilibria we call pattern formations. We prove their existence using Hamiltonian analysis, and present, numerically, a gallery of these stationary pattern formations.

Keywords: N-vortex problem; vortex dynamics; vortex filaments; Lie group analysis; symmetry analysis; equilibrium solutions; pattern formations.

DOI: 10.1504/IJANS.2014.068263

International Journal of Applied Nonlinear Science, 2014 Vol.1 No.4, pp.312 - 353

Received: 24 Jul 2013
Accepted: 18 Sep 2014
Published online: 08 Apr 2015 *

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