Integrable Chern-Simons Gauge Field Theory in 2+1 Dimensions

The classical spin model in planar condensed media is represented as the U(1) Chern-Simons gauge field theory. When the vorticity of the continuous flow of the media coincides with the statistical magnetic field, which is necessary for the model's integrability, the theory admits zero curvature connection. This allows me to formulate the model in terms of gauge - invariant fields whose evolution is described by the Davey-Stewartson (DS) equations. The Self-dual Chern-Simons solitons described by the Liouville equation are subjected to corresponding integrable dynamics. As a by-product the 2+1-dimensional zero-curvature representation for the DS equation is obtained as well as the new reduction conditions related to the DS-I case. Some possible applications for the statistical transmutation in the anyon superfluid and TQFT are briefly discussed.Comment: 16 pages, plain Te

Special functions with mod n symmetry and kaleidoscope of quantum coherent states

The set of mod $n$ functions associated with primitive roots of unity and discrete Fourier transform is introduced. These functions naturally appear in description of superposition of coherent states related with regular polygon, which we call kaleidoscope of quantum coherent states. Displacement operators for kaleidoscope states are obtained by mod $n$ exponential functions with operator argument and non-commutative addition formulas. Normalization constants, average number of photons, Heinsenberg uncertainty relations and coordinate representation of wave functions with mod n symmetry are expressed in a compact form by these functions.Comment: 12 pages, 4 figures, talk in The 32nd International Colloquium on Group Theoretical Methods in Physics (Group32), Prague, Czech Republic, 9-13 July 201

The Lax Pair by Dimensional Reduction of Chern-Simons Gauge Theory

We show that the Nonlinear Schr\"odinger Equation and the related Lax pair in 1+1 dimensions can be derived from 2+1 dimensional Chern-Simons Topological Gauge Theory. The spectral parameter, a main object for the Loop algebra structure and the Inverse Spectral Transform, has appear as a homogeneous part (condensate) of the statistical gauge field, connected with the compactified extra space coordinate. In terms of solitons, a natural interpretation for the one-dimensional analog of Chern-Simons Gauss law is given.Comment: 27 pages, Plain Te

q-Shock Soliton Evolution

By generating function based on the Jackson's q-exponential function and standard exponential function, we introduce a new q-analogue of Hermite and Kampe-de Feriet polynomials. In contrast to standard Hermite polynomials, with triple recurrence relation, our polynomials satisfy multiple term recurrence relation, derived by the q-logarithmic function. It allow us to introduce the q-Heat equation with standard time evolution and the q-deformed space derivative. We found solution of this equation in terms of q-Kampe-de Feriet polynomials with arbitrary number of moving zeros, and solved the initial value problem in operator form. By q-analog of the Cole-Hopf transformation we find a new q-deformed Burgers type nonlinear equation with cubic nonlinearity. Regular everywhere single and multiple q-Shock soliton solutions and their time evolution are studied. A novel, self-similarity property of these q-shock solitons is found. The results are extended to the time dependent q-Schr\"{o}dinger equation and the q-Madelung fluid type representation is derived.Comment: 15 pages, 6 figure