Self-Dual Vortices in Chern–Simons Hydrodynamics

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