Gauge transformation of the third kind for U(1)-invariant coupled Schroedinger equations

We consider a large class of canonical coupled nonlinear Schr¨odinger equations invariant over the action of the U(1)-group. The most general nonlinearity is taken into account through a matrix that, without loss of generality, can be separated into the sum of a Hermitian matrix and an anti-Hermitian matrix. The U(1)-symmetry implies the existence of a set of continuity equations for the conserved densities, where the corresponding currents have, in general, a nonlinear structure. For this class of coupled Schr¨odinger equations we introduce a nonlinear gauge transformation which changes the nonlinear matrix into another one, purely Hermitian. Consequently, the currents are transformed in the standard bilinear form. Generalization to noncanonical systems is also discussed. Some examples are presented to illustrate the applicability of the method.