A nonlinear Klein-Gordon equation for relativistic superfluidity

Many neutron star features can be accurately modeled only if one assumes that a significant portion of the neutron star interior is in a superfluid state and if relativitic effects are considered, and possible solutions to the underlying mathematical models include vortex solutions. It was recently shown that vorticity in relativistic superfluids can be studied under the framework of a nonlinear Klein-Gordon (NLKG) model in general curvilinear coordinates where the phase dynamics of solutions to this equation give rise to superfluidity (Xiong et al 2014 Phys. Rev. D 90 125019), and some numerical solutions were obtained. The aim of this paper will be to extract asymptotic solutions to obtain a better qualitative understanding of the possible relativistic superfluid dynamics possible under the NLKG model. We obtain asymptotic results for both spherically symmetric and cylindrically symmetric solutions, demonstrating that the solutions actually appear more regular in the relativistic regime compared to the non-relativistic limit. In fact, the asymptotic and numerical solutions actually show the best agreement in the relativistic case. We demonstrate that the relativistic effects actually tend to regularize or stabilize the solutions, relative to the non-relativistic solutions, which is an interesting finding. We then obtain a Thomas-Fermi-like perturbation result in the very large-mass limit where the kinetics become negligible relative to the self-interaction term (at leading order). We finally extend the NLKG model by assuming a curved spacetime with a metric generally used to model the space surrounding a neutron star, which is a novel generalization of the NLKG model to curved spacetime. We again obtain solutions in the large-mass limit for this case, and find that for such a spacetime non-stationary states (rather than simply stationary states) are possible in the large-mass limit.