Partially coherent solitons of variable shape in a slow Kerr-like medium: Exact solutions

We carry out a theoretical investigation of the properties of partially coherent solitons for media which have a slow Kerr-like nonlinearity. We find exact solutions of the $N\mathrm{th}$-order Manakov equations in a general form. These describe partially coherent solitons (PCSs) and their collisions. In fact, the exact solutions allow us to analyze important properties of PCSs such as stationary profiles of the spatial beams and effects resulting from their collisions. In particular, we find, analytically, the number of parameters that control the soliton shape. We present profiles which are symmetric as well as those which are asymmetric. We also find that collisions allow the profiles to remain stationary but cause their shapes to change.