Dynamics of energy critical wave equations

It is an important and largely open question to understand global behavior of nonlinear dispersive equations, when there are solitary waves. Most results are restricted to perturbative regimes. Recently there has been significant advances in energy critical wave equations. Here, the mechanism for the solution to split into dispersion and solitary waves at large times is better understood. I will explain in some details the main ingredients, including monotonicity formulae, and channel of energy inequalities (first introduced by Duyckaerts-Kenig-Merle).