Asymptotics for 1D Klein-Gordon Equations with Variable Coefficient Quadratic Nonlinearities

We initiate the study of asymptotic behavior 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 and temporal oscillations solutions. In case novel type modified scattering occurs that exhibits logarithmic slow-down decay rate along certain rays. non-resonant we introduce new normal form establish sharp estimates asymptotics presence critically dispersing constant cubic nonlinearity. models considered paper are motivated by stability kink classical nonlinear scalar field on real line.