in the case of a flat extra dimension, we were able to prove that any kink-like solution with nodes in the bulk is unstable. Thus we have focused on nodeless kink solutions and the trivial solution. However, in contrast to the flat case, in the presence of warping we were only able to find a sufficient condition for determing the stability of these solutions. We were therfore unable to analytically determine stability for nontrivial solutions in a warped background, even when that background is fixed (e.g., in the Randall-Sundrum model with no backreaction). Instead we were forced to determine stability numerically. Including the dynamics of the gravitational sector forces the inclusion of additional scalar fields whose purpose is to stabilize the size of the extra dimension. In that case we were again able to find nontrivial kink-like configurations, except now for a coupled multiple-field system. We have described a general graphical technique to find all possible static config- urations of the background equations with one kink scalar field and an arbitrary number of additional “stabilization” fields. The technique amounts to generating solution surfaces by varying the shooting parameters needed to solve the coupled system of equations. This technique also allows us to look for multiple solutions with the same action. We have demon- strated how to implement this technique in two simple examples, where we considered one kink field and one stabilization field in the presence of gravity. As in the flat case, when the potential for the kink field is a higher-order polynomial (leading to higher-order non- linearity in the field equations), we found that multiple solutions may exist for the same action. Interestingly, however, we also found multiple solutions for the same action when the kink potential was a fourth-order polynomial, which differs from the result obtained in a flat background. We have addressed the issue of stability only partially. We have derived the full 5D perturbative equations, including gravitational perturbations, for multiple scalar fields in the presence of a warped extra dimension. The system of equations constitute a matrix eigenvalue problem, which must be analyzed using an extension of the usual theorems coming from oscillation theory or Sturm-Liouville eigenvalue problems. Such techniques exist in the mathematical literature but due to the complexity of the task, we have left the numerical analysis of the general case for a later work. 25