We give a rigorous proof for the linear stability of the Skyrmion. In addition, we provide new proofs for the existence of the Skyrmion and the GGMT bound.
Due to the complexity of the field equation, there are not many rigorous results on dynamical aspects of the Skyrme model. In @cite_17 , small data global well-posedness and scattering is proved and @cite_25 establishes large-data global well-posedness. There is also some recent activity on the related but simpler Adkins-Nappi model, see e.g. @cite_13 @cite_20 @cite_18 . From a numerical point of view, the linear stability of the Skyrmion is addressed in @cite_6 and @cite_7 studies the nonlinear stability. As far as the method of proof is concerned, we note that our approach is in parts inspired by @cite_24 .
@cite_18 @cite_7 @cite_6 @cite_24 @cite_13 @cite_25 @cite_20 @cite_17
1989169947 1510406861 2962852827 2048738839 2104249908 2593560900 2161356684 2004482079
Abstract We investigate the stability of our recently constructed self-gravitating skyrmions in the framework of small time-dependent perturbations. It is shown that the frequency spectrum for radial perturbations of the coupled Einstein-Skyrme equations can be reduced to the energy spectrum of a p-wave Schrodinger equation with a bounded effective potential, which is determined by the equilibrium solution. Bound states of this Schrodinger equation correspond to exponentially growing modes. It turns out that there are no such modes, as long as the relevant dimensionless coupling constant ( κ ) is less than a critical value, κ c . For larger values of κ there is exactly one unstable mode. Therefore, the Skyrme “stars” become unstable (in the sense of Liapunov) for κ > κ c , in spite of the fact that their topological winding number is conserved. The linear stability for κ κ c (with respect to radial modes) is, of course, only a necessary condition for (non-linear) stability. Similar results are expected to hold for our new black hole solutions of the Einstein-Skyrme system. We consider the dynamics of hedgehog solutions in the @math -dimensional Skyrme model which is an energy-supercritical problem. We introduce a new strategy to prove global wellposedness for arbitrarily large initial data. We study a generalization of energy super-critical wave maps due to Adkins and Nappi that can also be viewed as a simplified version of the Skyrme model. These are maps from 1 + 3 dimensional Minkowski space that take values in the 3-sphere, and it follows that every finite energy Adkins–Nappi wave map has a fixed topological degree which is an integer. Here we initiate the study of the large data dynamics for Adkins–Nappi wave maps by proving that there is no type II blow-up in the class of maps with topological degree zero. In particular, any degree zero map whose critical norm stays bounded must be global-in-time and scatter to zero as ( t ). Consider a system governed by the time-dependent Schrodinger equation initialized in its ground state. When subjected to weak (size e) parametric forcing by an “ionizing field” (time-varying), the state decays with advancing time due to coupling of the bound state to radiation modes. The decay rate of this metastable state is governed by Fermi’s golden rule, Γ[V], which depends on the potential V and the details of the forcing. We pose the following potential design problem: find Vopt which minimizes Γ[V] (maximizes the lifetime of the state) over an admissible class of potentials with fixed spatial support. We formulate this problem as a constrained optimization problem and prove that an admissible optimal solution exists. Then, using quasi-Newton methods, we compute locally optimal potentials. These have the structure of a truncated periodic potential with a localized defect. In contrast to optimal structures for other spectral optimization problems, our optimizing potentials appear to be interior point... Linear perturbations of black hole models by a variety of fields are considered. Perturbing fields include the zero rest mass scalar field in the case of Reissner-Nordstrom, and gravitational, electromagnetic and zero rest mass scalar perturbation in the case of the Kerr model. The analysis deals with the Ψ 0 components (in the Newman-Penrose (1962) formalism) of non-zero spin fields. The symmetry properties of the models are used to derive the crucial condition th at the field be singular on the inner horizon. This condition is independent of the field propagation equation. Initial data are then given in terms of incoming radiation from f - is shown that there exist wellbehaved initial data sets for which the resultant fields are singular on the inner horizon. It is emphasized that this instability result is dependent only on the global symmetries and causal structure of the models considered, and is independent of the precise nature of the perturbing field. This paper contains the second part of a two-part series on the stability and instability of extreme Reissner–Nordstrom spacetimes for linear scalar perturbations. We continue our study of solutions to the linear wave equation ( g =0 ) on a suitable globally hyperbolic subset of such a spacetime, arising from regular initial data prescribed on a Cauchy hypersurface Σ0 crossing the future event horizon ( H ^ + ). We here obtain definitive energy and pointwise decay, non-decay and blow-up results. Our estimates hold up to and including the horizon ( H ^ + ). A hierarchy of conservations laws on degenerate horizons is also derived. This paper is part of the radial asymptotic stability analysis of the ground state soliton for either the cubic nonlinear Schrodinger or Klein–Gordon equations in three dimensions. We demonstrate by a rigorous method that the linearized scalar operators which arise in this setting, traditionally denoted by L±, satisfy the gap property, at least over the radial functions. This means that the interval (0, 1] does not contain any eigenvalues of L± and that the threshold 1 is neither an eigenvalue nor a resonance. The gap property is required in order to prove scattering to the ground states for solutions starting on the centre-stable manifold associated with these states. This paper therefore provides the final instalment in the proof of this scattering property for the cubic Klein–Gordon equation in the radial case, see the recent theory of Nakanishi and the third author. The method developed here is quite general, and applicable to other spectral problems which arise in the theory of nonlinear equations. We consider a spherically symmetric, double characteristic initial value problem for the (real) Einstein-Maxwell-scalar field equations. On the initial outgoing characteristic, the data is assumed to satisfy the Price law decay widely believed to hold on an event horizon arising from the collapse of an asymptotically flat Cauchy surface. We establish that the heuristic mass inflation scenario put forth by Israel and Poisson is mathematically correct in the context of this initial value problem. In particular, the maximal future development has a future boundary over which the space-time is extendible as a C0 metric but along which the Hawking mass blows up identically; thus, the space-time is inextendible as a C1 metric. In view of recent results of the author in collaboration with I. Rodnianski, which rigorously establish the validity of Price's law as an upper bound for the decay of scalar field hair, the C0 extendibility result applies to the collapse of complete, asymptotically flat, spacelike initial data where the scalar field is compactly supported. This shows that under Christodoulou's C0 formulation, the strong cosmic censorship conjecture is false for this system. © 2005 Wiley Periodicals, Inc.