Nonlinear Long-Range Resonant Scattering and Kink Dynamics

We study the nonlinear Klein-Gordon equation, in one dimension, with a qudratic term and variable coefficient qubic term. This equation arises from the asymptotic stability theory of the kink solution.Our main result is the global existence and decay estimates for this equation. We discovered a striking new phenomena in this problem: a resonant interaction between the spacial frequencies of the nonlinear coefficient and the temporal oscillations of the solution.
The proof requires the development of new normal-form transformations. This is a joint work with H. Lindblad, I. Rodnianski and J. Sterbenz.