Scattering for the quadratic Klein-Gordon equation with inverse-square potential and related problems

We study the scattering of certain non-linear dispersive partial differential equations (PDEs). We may interpret a solution to such a PDE as a model starting off at a given initial state and being evolved according to the PDE. In a linear setting, such a model would disperse its energy across physical space as it evolves in time. We investigate settings for the initial state such that the (oftentimes difficult to understand) nonlinear evolution of the model in fact displays this linear behaviour after a large amount of time.