Bifurcation of soliton bags: Two-dimensional linear σ model

We investigate classical solutions of the two-dimensional linear σ model, including solutions on a finite domain. Such solutions arise when nucleons, modelled by solitons, are bound together to form a nucleus in which they are confined to a domain of the order of the soliton size. We investigate the behaviour of the solitons as the domain radius is varied, and find that for a small radius only a constant solution is allowed, but that at a critical radius a nontrivial soliton solution bifurcates from this constant solution. In the quantum theory this can be viewed as the transition from a uniform plasma to a nucleon state. We find exact solutions of the nonlinear equations, including a new kink solution with a nontrivial fermion interaction, for which this behaviour can be demonstrated explicitly. More generally, we detect the presence of a bifurcation by linear analysis, which also allows the calculation of the exact bifurcation radius and corresponding energy. The linear analysis is carried out for several sets of boundary conditions, all of which lead to periodic soliton solutions for general values of the coupling parameters.