Compactons and semi-compactons in the extreme baby Skyrme model

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.