Topological solutions in the self-dual Chern-Simons theory: existence and approximation

Abstract

In this paper a globally convergent computational scheme is established to approximate a topological multivortex solution in the recently discovered self-dual Chern-Simons theory in R2. Our method which is constructive and numerically efficient finds the most superconducting solution in the sense that its Higgs field has the largest possible magnitude. The method consists of two steps: first one obtains by a convergent monotone iterative algorithm a suitable solution of the bounded domain equations and then one takes the large domain limit and approximates the full piane solutions. It is shown that with a special choice of the initial guess function, the approximation sequence approaches exponentially fast a solution in R2. The convergence rate implies that the truncation errors away from local regions are insignificant.