We consider various examples of critical nonlinear partial differential equations which have the following common features: they are Hamiltonian, of dispersive nature, have a conservation law invariant by scaling, and have solutions of nonlinear type (their asymptotic behavior in time differs from the behavior of solutions of linear equations). The main questions concern the possible behaviors one can expect asymptotically in time. Are there many possibilities, or on the contrary very few universal behaviors depending on the type of initial data? We shall see that the asymptotic behavior of solutions starting with general or constrained initial data is related to very few special solutions of the equation. This will be illustrated through different examples related to classical problems.