Asymptotic stability of the Skyrmion

Abstract

We study the asymptotic behavior of spherically symmetric solutions in the Skyrme model. We show that the relaxation to the degree-one soliton (called the Skyrmion) has a universal form of a superposition of two effects: exponentially damped oscillations (the quasinormal ringing) and a power law decay (the tail). The quasinormal ringing, which dominates the dynamics for intermediate times, is a linear resonance effect. In contrast, the polynomial tail, which becomes uncovered at late times, is shown to be a nonlinear phenomenon.Introduction. Stable stationary solutions are natural candidates for the endstates of evolution of many physical systems. The relaxation to these states is well understood for dissipative systems, described by diffusion equations, however for conservative Hamiltonian systems on unbounded domains the problem is much more difficult because there is no local dissipation of energy and convergence to equilibrium is due to dispersion, that is, radiation of excess energy to infinity [1]. Understanding dissipation by dispersion is important physically because the radiation emitted during the approach to equilibrium encodes information about the attractor -this kind of inverse problem has various applications, for instance in identifying black holes via gravitation radiation emitted during the last stages of gravitational collapse.In this paper we address the problem of relaxation to equilibrium in a very simple setting of the spherically symmetric Skyrme model [2]. This model, apart from its physical relevance in particle physics, is attractive theoretically because the equilibrium state is completely rigid: it has no moduli and no internal degrees of freedom, which makes the mathematical analysis feasible. Yet, despite the simplicity of the model, the relaxation process exhibits a surprising feature: after a transient oscillatory exponential decay (so called quasinormal ringing) which is a linear resonance effect, there proceeds a power law tail which has a nonlinear origin. Pointing out the failure of the linear perturbation theory in capturing the asymptotic dynamics is the main message of this paper. Below, after introducing the model, we first describe the quasinormal ringing using the linear perturbation theory, then we present the numerical evidence for the asymptotic behavior of solutions, and finally we explain the tail using the nonlinear perturbation theory.Background. Let M be a spacetime with a metric η µν and N be a complete Riemannian manifold with a metric g AB . Consider a map U : M → N and denote by S µν = g AB ∂ µ U A ∂ ν U B the pulled back metric. The (generalized) Skyrme model is defined by the lagrangian

Radial vibrations of BPS skyrmions

“…The quasi-normal modes behave like: The gray areas represent the non-propagating region below the mass threshold. a) = 0.2 oscillational mode excited, almost constant amplitude, small radiation via second harmonics; b) = 0 pure sG case, slow modes on the threshold ω = 1 dominate long time evolution; c) = −0.1, fast decay of the resonance mode and remaining tail at the mass threshold; d) similar to c) but longer decay of the resonance and clearly seen difference in frequency e) = −0.25 massless case with decaying resonance and polynomial tail described by Bizoń [39].…”

“…Radial excitations may be studied using various methods with different levels of precision and rigor [34] - [39]. The simplest description assumes that the excitation may be described by a uniform rescaling (Derrick scaling) of the field variables, U (t, x) = U 0 ( x/ρ(t)), where U 0 is the static skyrmion solution.…”

Late Miocene/Early Pliocene vertebrate fauna from Mallorca (Balearic Islands, Western Mediterranean): an update

“…The deposit, located 5 m asl, consists of consolidated red silt of karstic origin which are the floor sediment of a collapsed cave in the Upper Miocene limestone of the so-called Reef Complex (Pomar et al 1990;Gómez-Pujol et al 2007) of Late Tortonian-Messinian chronology (Bizon et al 1973). Just a small portion of the sediments (around 1 m 2 at the northeast extreme of the cave sediments) has yielded fossil remains while the rest of the sediment seems to be sterile.…”

Large data global regularity for the classical equivariant Skyrme model

“…Then there exists a global radial solution u to the Cauchy problem associated to (2) with (u(0), u t (0)) = (u 0 , u 1 ), satisfying (4) and which is of semilinear type and looks considerably simpler than (2). Surprisingly, Shatah [18] showed that there are smooth data that lead to solutions of this equation that blow up in finite time, with Turok and Spergel [23] later finding such solutions in closed form.…”

“…As these are evolution equations, one natural question to study about them is the well-posedness of the associated initial value problem. This is a very challenging task, mainly due to the quasilinear nature of the equation (displayed above by (2)) and the fact that scaling heuristics in a small energy scenario show that the Cauchy problem is supercritical with respect to the energy (for more details, see [8]).…”