I will discuss classical and recent results regarding asymptotic stability of solitary waves i.e., localized solutions of nonlinear wave equations propagating without changing shape. Begining with the work of Soffer and Weinstein in the ’90 we know that, under certain assumptions, solutions starting close to a solitary wave shadow nearby solitary waves before collapsing on one. The mathematical analysis relies on using dispersive estimates for the linearized dynamics at a fixed (rather arbitrarily chosen) solitary wave to control the nonlinearity. I will show that the assumptions on the nonlinearity can be significantly relaxed provided one uses a time dependent linearization which follows the different solitary waves shadowed by the actual solution. Of course, this new perspective requires new dispersive estimates for time dependent wave type operators and I will discuss how they can be obtained. This is joint work with A. Zarnescu (U. Sussex, England), O. Mizrak (Mersin U., Turkey), and R. Skulkhu (Mahidol U., Thailand).