Self-dual Vortices in Chern-simons Hydrodynamics

The classical theory of non-relativistic charged particle interacting with U(1) gauge field is reformulated as the Schrödinger wave equation modified by the de-Broglie-Bohm quantum potential nonlinearity. For, (1 h̄) deformed strength of quantum potential the model is gauge equivalent to the standard Schrödinger equation with Planck constant h̄, while for the strength (1 + h̄), to the pair of diffusion-anti-diffusion equations. Specifying the gauge field as Abelian Chern-Simons (CS) one in 2+1 dimensions interacting with the Nonlinear Schrödinger field (the Jackiw-Pi model), we represent the theory as a planar Madelung fluid, where the Chern-Simons Gauss law has simple physical meaning of creation the local vorticity for the fluid flow. For the static flow, when velocity of the center-ofmass motion (the classical velocity) is equal to the quantum one (generated by quantum potential velocity of the internal motion), the fluid admits N-vortex solution. Applying the Auberson-Sabatier type gauge transform to phase of the vortex wave function we show that deformation parameter h̄, the CS coupling constant and the quantum potential strength are quantized. Reductions of the model to 1+1 dimensions, leading to modified NLS and DNLS equations with resonance soliton interactions are discussed.