  We study self-similar solutions of NLS-type dynamical systems. Lagrangian
approach is used to show that they can be reduced to three canonical forms,
which are related by Miura transformations. The fourth Painleve equation (PIV)
is central in our consideration - it connects Heisenberg model, Volterra model
and Toda model to each other. The connection between the rational solutions of
PIV and Coulomb gas in a parabolic potential is established. We discuss also
the possibility to obtain an exact solution for optical soliton i.e. of the NLS
equation with time-dependent dispersion.
