Global internal mode dynamics for the 1D quartic Klein-Gordon equation

We study the quartic Klein–Gordon equations with a potential in one space dimension as model problems and establish a global description of internal mode dynamics and radiation damping. It is well-known that internal modes not only decay slowly, but their amplitudes can also lose smallness at large times; combined with the slow dispersive decay in $1$ dimension, this makes the global analysis particularly delicate. Our approach is based on distorted Fourier transforms, normal form transformations, together with a collection of refined dispersive decay and smoothing estimates, which we exploit even at the level of the internal mode equation itself. This a joint work with Gael  Diebou, Adilbek Kairzhan and Fabio Pusateri.