The self-focusing and splitting mechanisms of waves governed by the cubic nonlinear Schrodinger equation with anisotropic dispersion are investigated numerically by means of an adaptive mesh refinement code. Wave-packets having a power far above the self-focusing threshold undergo a transversal compression and are shown to split into two symmetric peaks. These peaks can sequentially decay into smaller-scale structures developing near the front edge of a shock, as long as their individual power remains above threshold, until the final dispersion of the wave. Their phase and amplitude dynamics are detailed and compared with those characterizing collapsing objects with no anisotropic dispersion. Their ability to mutually coalesce is also analyzed and modeled from the interaction of Gaussian components. Next, bunch-type and snake-type instabilities, which result from periodic modulations driven by even and odd localized modes, are studied. The influence of the initial wave amplitude, the amplitude and wavenumber of the perturbations on the interplay of snake and bunch patterns are finally discussed. (C) 2001 Elsevier Science B.V. All rights reserved.