The complex Sine-Gordon equation as a symmetry flow of the AKNS Hierarchy

It is shown how the complex sine-Gordon equation arises as a symmetry flow of the AKNS hierarchy. The AKNS hierarchy is extended by the ``negative'' symmetry flows forming the Borel loop algebra. The complex sine-Gordon and the vector Nonlinear Schrodinger equations appear as lowest negative and second positive flows within the extended hierarchy. This is fully analogous to the well-known connection between the sine-Gordon and mKdV equations within the extended mKdV hierarchy. A general formalism for a Toda-like symmetry occupying the ``negative'' sector of sl(N) constrained KP hierarchy and giving rise to the negative Borel sl(N) loop algebra is indicated.Comment: 8 pages, LaTeX, typos corrected, references update

Noncommutativity due to spin

Using the Berezin-Marinov pseudoclassical formulation of spin particle we propose a classical model of spin noncommutativity. In the nonrelativistic case, the Poisson brackets between the coordinates are proportional to the spin angular momentum. The quantization of the model leads to the noncommutativity with mixed spacial and spin degrees of freedom. A modified Pauli equation, describing a spin half particle in an external e.m. field is obtained. We show that nonlocality caused by the spin noncommutativity depends on the spin of the particle; for spin zero, nonlocality does not appear, for spin half, $\Delta x\Delta y\geq\theta^{2}/2$, etc. In the relativistic case the noncommutative Dirac equation was derived. For that we introduce a new star product. The advantage of our model is that in spite of the presence of noncommutativity and nonlocality, it is Lorentz invariant. Also, in the quasiclassical approximation it gives noncommutativity with a nilpotent parameter.Comment: 11 pages, references adda

Dressing approach to the nonvanishing boundary value problem for the AKNS hierarchy

We propose an approach to the nonvanishing boundary value problem for integrable hierarchies based on the dressing method. Then we apply the method to the AKNS hierarchy. The solutions are found by introducing appropriate vertex operators that takes into account the boundary conditions.Comment: Published version Proc. Quantum Theory and Symmetries 7 (QTS7)(Prague, Czech Republic, 2011