Quasi-periodic and periodic solutions for coupled nonlinear Schrodinger equations of Manakov type

We consider travelling periodic and quasi-periodic wave solutions in coupled nonlinear Schrodinger equations. In fibre optics these equations can be used to model single-mode fibres with strong birefringence, and two-mode optical fibres. Recently these equations appear as a model describing pulse-pulse interactions in wavelength-division-multiplexed channels of optical fibre transmission systems. In some cases this model reduces to the integrable Manakov system (IMS). Two-phase quasi-periodic solutions for the IMS are given in terms of two-dimensional Kleinian functions. The reduction of quasi-periodic solutions to elliptic functions is discussed. New solutions are found in terms of generalized Hermite polynomials, which are associated with two-gap Treibich-Verdier potentials.