With the advent of computers in the middle of the 20’th century, through the remarkable computations of Fermi-Pasta-Ulam (mid 50s) and Kruskal-Zabusky (mid 60s) it was observed numerically that nonlinear equations modeling wave propagation asymptotically exhibit as superposition of “traveling waves” and “radiation”. This has become known as the “soliton resolution conjecture”. This conjecture, roughly speaking, says that any reasonable solution to a disperive equation eventually resolves into a superposition of a radiation component plus a finite number of "nonlinear bound states" or "solitons".  After an informal introduction to dispersive equations, I will discuss the soliton resolution and long-time dynamics for the modified Korteweg-de vries (mKdV) equation with initial conditions in some weighted Sobolev spaces combing ideas from integrable systems and partial differential equations (PDEs).