  We study the existence and uniqueness of solutions to the vector field
Peierls-Nabarro model for curved dislocations in a transversely isotropic
medium. Under suitable assumptions for the misfit potential on the slip plane,
we reduce the 3D Peierls-Nabarro model to a nonlocal scalar Ginzburg-Landau
equation. For a particular range of elastic coefficients, the nonlocal scalar
equation with explicit nonlocal positive kernel is derived. We prove that any
stable steady solution has a one-dimensional profile. As a result, we obtain
that solutions to the scalar equation, as well as the original 3D system, are
characterized as a one-parameter family of straight dislocations. This paper
generalizes results found previously for the full isotropic case to an
anisotropic setting.
