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Register a new userOn peakons and Q-balls in the baby Skyrme model
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Jakub Lis
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In this paper we investigate the Q-ball Ansatz in the baby Skyrme model. First, the appearance of peakons, i.e. solutions with extremely large absolute values of the second derivative at maxima, is analyzed. It is argued that such solutions are intrinsic to the baby Skyrme model and do not depend on the detailed form of a potential used in calculations. Next, we concentrate on compact non spinning Q-balls. We show the failure of a small parameter expansion in this case. Finally, we explore the existence and parameter dependence of Q-ball solutions.
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The static baby Skyrme model is investigated in the extreme limit where the energy functional contains only the potential and Skyrme terms, but not the Dirichlet energy term. It is shown that the model with potential $V=frac12(1+phi_3)^2$ possesses solutions with extremely unusual localization properties, which we call semi-compactons. These minimize energy in the degree 1 homotopy class, have support contained in a semi-infinite rectangular strip, and decay along the length of the strip as $x^{-log x}$. By gluing together several semi-compactons, it is shown that every homotopy class has linearly stable solutions of arbitrarily high, but quantized, energy. For various other choices of potential, compactons are constructed with support in a closed disk, or in a closed annulus. In the latter case, one can construct higher winding compactons, and complicated superpositions in which several closed string-like compactons are nested within one another. The constructions make heavy use of the invariance of the model under area-preserving diffeomorphisms, and of a topological lower energy bound, both of which are established in a general geometric setting. All the solutions presented are classical, that is, they are (at least) twice continuously differentiable and satisfy the Euler-Lagrange equation of the model everywhere.
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Non-topological solitons such as Q-balls and Q-shells have been studied for scalar fields invariant under global and gauged U(1) symmetries. We generalize this framework to include a Proca mass for the gauge boson, which can arise either from spontaneous symmetry breaking or via the Stuckelberg mechanism. A heavy (light) gauge boson leads to solitons reminiscent of the global (gauged) case, but for intermediate values these Proca solitons exhibit completely novel features such as disconnected regions of viable parameter space and Q-shells with unbounded radius. We provide numerical solutions and excellent analytic approximations for both Proca Q-balls and Q-shells. These allow us to not only demonstrate the novel features numerically, but also understand and predict their origin analytically.
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