The asymptotic stability analysis of topological solitons such as the well-known static ``kink'' solution of the \phi^4 model in one space dimension requires an understanding of the asymptotic behavior of small solutions to 1D Klein-Gordon equations with variable coefficient quadratic and cubic nonlinearities. In this talk I will first describe the challenges for the scattering theory of dealing with the long-range nature of such nonlinearities and I will explain the difficulties caused by variable coefficients. Then I will present a new result on decay estimates and asymptotics for small solutions to 1D Klein-Gordon equations with constant and variable coefficient cubic nonlinearities. The main novelty of our approach is the use of pointwise-in-time local decay estimates to deal with the variable coefficient nonlinearity. This is joint work with Hans Lindblad and Avy Soffer.