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The asymptotic stability analysis of one-dimensional topological solitons such as the well-known â kinkâ in the \$phi^4$ model 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 difficulties caused by variable coefficients to deal with the long-range nature of such nonlinearities. Then I will present a new result on sharp 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. If time permits, I will also discuss work in progress on the variable coefficient quadratic case, which exhibits a striking resonant interaction between the spatial oscillations of the variable coefficient and the temporal oscillations of the solutions. This is joint work with Hans Lindblad and Avy Soffer.