Self-consistent turbulence in the two-dimensional nonlinear Schrodinger equation with a repulsive potential.

The dynamics of dark solitons (vortices) with the same topological charge ( vorticity) in the two-dimensional nonlinear Schrodinger (NLS) equation in a defocusing medium is studied. The dynamics differ from those in incompress ible media due to the possibility of energy and angular momentum radiation. The problem of the breakup of a multicharged dark soliton, which is a loca l decrease of the wave function intensity, into a number of chaotically mov ing vortices with single charge, is studied both analytically and numerical ly. After an initial period of intensive wave radiation, there emerges a no nuniform, steady turbulent self-organized motion of these vortices which is restricted in space by the size of the potential well of the initial multi charged dark soliton. Separate orbits of finite widths arise in this turbul ent motion. That is, the statistical probability to observe a vortex in a g iven point has maxima near certain points (orbit positions). In spite-of th e fact that numerical calculations were performed in a finite region, the t urbulent distributions of the Vortices dd not depend on the size of the con tainer when its radius is larger than the size of the potential well of the primary multicharged dark soliton. The steady turbulent distribution of Vo rtices on these orbits can be obtained as; the extremal of the Lyapunov fun ctional of the NLS equation,and obeys some simple rules. The first is the a bsence of Cherenkov resonance with linear (sound) waves. The second is the condition of a potential energy maximum in the region of vortex motion. The se conditions give an approximately equidistant disposition of orbits of th e same number of vortices on each orbit, which corresponds to a constant ro tating Velocity. The magnitude of this velocity is mainly determined by the sound velocity. An integral estimation of the self-consistent rotation of the vortex zone is given. [S1063-651X(99)08906-0].