Standing waves in a counter-rotating vortex filament pair

Abstract

The distance among two counter-rotating vortex filaments satisfies a beam-type of equation according to the model derived in [15]. This equation has an explicit solution where two straight filaments travel with constant speed at a constant distance. The boundary condition of the filaments is 2π-periodic. Using the distance of the filaments as bifurcating parameter, an infinite number of branches of periodic standing waves bifurcate from this initial configuration with constant rational frequency along each branch.MSC: 35B10, 35B32

Free vibrations in a wave equation modeling MEMS

“…The choice of the maximal p 0 is equivalent to the choice of the minimal period T = 2πq/p 0 . This argument is similar to the argument used in [15], [17] and [16].…”

“…The main challenge encountered when proving the existence of the periodic solutions is that the trivial branch u λ and its associated spectrum are not known explicitly. Indeed, while the existence of continuous families of periodic solutions has been obtained before for Hamiltonian PDEs in [14], [15], [16] and [17], in those articles the trivial branch and the spectrum of its linearized equation are known explicitly. We overcome our problem with delicate estimates of the spectrum which depend on an accurate estimate for the steady state u λ * at the critical value λ * .…”

“…The small divisor problem was avoided in [13], [10] and [14] by imposing restrictions on the temporal period of the solutions of a nonlinear wave equation. By imposing similar restrictions, the articles [15] obtains the existence of continuous branches of periodic solutions for a nonlinear wave equation in a sphere, and in [16] and [17] for a Hamiltonian PDE appearing in the n-vortex filament problem.…”

Standing waves of fixed period for $n+1$ vortex filaments

“…our result can be obtained only in a narrow set of parameters where L(ω 0 ) has a nontrivial kernel. This is also the case of the counter-rotating vortex filament pair studied in [11], but this is the first time that periodic solutions without small divisors are obtained in a genuine non-linear Hamiltonian PDE using this method.…”

“…, n that describe the positions of n + 1 vertically oriented vortex filaments. Different aspects of this problem have been investigated in [3,4,8,11,12,13,17] and references therein. In this article we study central configurations of n + 1 vortex filaments with n filaments of equal circulation and one filament of opposite circulation.…”