The classical theory of non-relativistic charged particle interacting with
U(1) gauge field is reformulated as the Schr\"odinger wave equation modified by
the de-Broglie-Bohm quantum potential nonlinearity. For, (1 - ℏ2)
deformed strength of quantum potential the model is gauge equivalent to the
standard Schr\"odinger equation with Planck constant ℏ, while for the
strength (1 + ℏ2), 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 Schr\"odinger 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-of-mass 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 ℏ, 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.Comment: 12 pages, Tex, to be published in Proc. "NEEDS'2000", Gokova, Turkey,
2000; Theor. and Math.Physic