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

We consider travelling periodic and quasi–periodic wave solutions in coupled nonlinear Schrödinger 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.