  We initiate the study of the asymptotic behavior of small solutions to
one-dimensional Klein-Gordon equations with variable coefficient quadratic
nonlinearities. The main discovery in this work is a striking resonant
interaction between specific spatial frequencies of the variable coefficient
and the temporal oscillations of the solutions. In the resonant case a novel
type of modified scattering behavior occurs that exhibits a logarithmic
slow-down of the decay rate along certain rays. In the non-resonant case we
introduce a new variable coefficient quadratic normal form and establish sharp
decay estimates and asymptotics in the presence of a critically dispersing
constant coefficient cubic nonlinearity. The Klein-Gordon models considered in
this paper are motivated by the study of the asymptotic stability of kink
solutions to classical nonlinear scalar field equations on the real line.
