Nonlinear polarization dynamics. I. The single-pulse equations

We consider the polarization dynamics of a pulse propagating through an arbitrary nonlinear medium, in the limit of small nonlinearities, anisotropies, and dispersion, using the full SO(3) covariance of the Stokes parameters. The invariants of the motion are discovered and physically interpreted, a complete analogy with the ``airplane-and-rotor'' problem of rigid-body dynamics is established, and a full classification of the solutions for parity-invariant and non-parity-invariant media, with all propagation-axis rotation symmetries, is presented. In all cases, the problem is reduced to quadrature; in most cases, we find analytic solutions in terms of well-known functions.