Global continua of periodic solutions of singular first-order Hamiltonian systems of N-vortex type

Abstract

The paper deals with singular first order Hamiltonian systems of the formwhere J ∈ R 2×2 defines the standard symplectic structure in R 2 , and the Hamiltonian H is of N -vortex type:This is defined on the configuration space {(z 1 , . . . , z N ) ∈ Ω 2N : z j = z k for j = k} of N different points in the domain Ω ⊂ R 2 . The function F : Ω N → R may have additional singularities near the boundary of Ω N . We prove the existence of a global continuum of periodic solutions z(t) = (z 1 (t), . . . , z N (t)) ∈ Ω N that emanates, after introducing a suitable singular limit scaling, from a relative equilibrium Z(t) ∈ R 2N of the N -vortex problem in the whole plane (where F = 0). Examples for Z include Thomson's vortex configurations, or equilateral triangle solutions. The domain Ω need not be simply connected. A special feature is that the associated action integral is not defined on an open subset of the space of 2π-periodic H 1/2 functions, the natural form domain for first order Hamiltonian systems. This is a consequence of the singular character of the Hamiltonian. Our main tool in the proof is a degree for S 1 -equivariant gradient maps that we adapt to this class of potential operators.MSC 2010: Primary: 37J45; Secondary: 37N10, 76B47

Periodic solutions with prescribed minimal period of vortex type problems in domains

“…z . In [2,3,5] the authors also obtained periodic solutions converging towards z 0 . More precisely, they produced a family of T r -periodic solutions z (r) (t), parameterized over r ∈ (0, r 0 ) with |z (r) j (t) − z 0 | = r + o(r) and T r → 0 as r → 0.…”

“…It is worthwhile to mention that systems like (1.1) also arise in other contexts from mathematical physics, e.g. in models from superconductivity (Ginzburg-Landau-Schrödinger equation), or in equations modeling the dynamics of a magnetic vortex system in a thin ferromagnetic film (Landau-Lifshitz-Gilbert equation); see [5] for references to the literature. The domain can also be a subset of a two-dimensional surface.…”

The Morse property for functions of Kirchhoff-Routh path type

“…In [2,3] periodic solutions of the N-vortex problem from fluid dynamics have been found near stable critical points of the Robin function. If the critical point is non-degenerate then it has been proved in [1] that there exists a smooth one-parameter curve of periodic solutions.…”

Stability of Periodic Solutions of the N-vortex Problem in General Domains

“…We would also like to mention that for the existence alone, a 0 does not need to be a nondegenerate critical point, it is enough that a 0 is topological stable, i.e., that a 0 is an isolated critical point with nonvanishing Brouwer index, see [3,5]. But under these weaker assumption it is not clear, if the induced solutions form a continuous family of solutions, which is needed for our further stability analysis.…”