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
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...