The 2D Chern-Simons-Schr{ö}dinger system reduced to 1D

We study a mean-field model for a system of 2D abelian anyons, given by the dynamics of a Schr{ö}dinger matter field coupled to a Chern-Simons gauge field. We derive an effective 1D equation by adding a strongly anisotropic trapping potential (wave-guide) acting on the Schr{ö}dinger field, and tracing out the tight confinement direction. The effective dynamics in the loose direction of the wave-guide turns out to be governed by the classical 1D quintic NLS equation.