Many wave phenomena in nature, from water waves to light pulses in optical fibers, can be described by nonlinear dispersive equations. Among their solutions, solitons are particularly remarkable: they are localized waves that retain their shape as they travel, behaving almost like particles despite being waves.
In this talk, I will discuss recent progress on the asymptotic stability of topological solitons - such as kinks and vortices - in low-dimensional settings, where strong interactions between solitons and dispersive radiation give rise to especially rich and subtle dynamics.