Collapse transformation for self-focusing solitary waves in boundary-layer type shear flows

Evolution of long-wave weakly nonlinear two-dimensional perturbations in parallel boundary-layer type shear flows is considered within the model simplified by using the paraxial approximation. The transverse instability of 1D solitons is shown to result in formation of 2D collapsing clusters which evolve in a self-similar manner. The explicit “critical” collapse transformation of 2D solitons is found within the framework of the Whitham adiabatic approach.