Self-similar solutions and collective coordinate methods for nonlinear Schrِdinger equations

In this paper we study the phase of self-similar solutions to general nonlinear Schrِdinger equations. From this analysis we gain insight on the dynamics of non-trivial solutions and a deeper understanding of the way collective coordinate methods work. We also find general evolution equations for the most relevant dynamical parameter w(t) corresponding to the width of the solution. These equations are exact for self-similar solutions and provide a shortcut to find approximate evolution equations for the width of non-self-similar solutions similar to those of collective coordinate methods.