Symmetry analysis of equilibrium solutions for a pair of interacting vortex filaments (Q258015)

Summary: 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 \(\Gamma=\Gamma_1=\Gamma_2=1\), give rise to relative equilibria. We show, when \(\Gamma=\Gamma_1=-\Gamma_2=1\), 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.