On the Born-Infeld Abelian Higgs cosmic strings with symmetric vacuum

We study a system of elliptic equations from the Abelian BornInfeld system coupled with the Einstein equations under the boundary condition of the symmetric vacuum(nontopological type). When the total string number satisfies 1 ≤ N < 1 4πG , where G is the gravitational constant, we construct a family of solutions to the system. The qualitative properties of the solutions are quite different from the solutions with the boundary condition of the broken vacuum symmetry.