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

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.

Automated summary

This study investigates the asymptotic behavior of small solutions to one-dimensional Klein-Gordon equations with variable coefficient quadratic nonlinearities. It discovers a striking resonant interaction between specific spatial frequencies of the variable coefficient and the temporal oscillations of the solutions, leading to a novel type of modified scattering behavior. Sharp decay estimates and asymptotics are established in both resonant and non-resonant cases.