text: 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.
  @cite_18: 
  @cite_7: 
  @cite_6: 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.
  @cite_24: 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.
  @cite_13: We investigate a semilinear modification for the wave map prob- lem proposed by Adkins and Nappi (1), and prove that in the equivariant case the solution remain continuous at the first possible singularity. This is usually one of the steps in proving existence of global smooth solutions for certain equations.
  @cite_25: 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.
  @cite_20: 
  @cite_17: 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 ).
summary: 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 .