We consider optical pulse propagation in one spatial direction and observe that for lossless media, the resulting Maxwell equations are of the form of an infinite dimensional Hamiltonian system evolving in the spatial direction. A simplified uni-directional model is derived for waves running mainly in one direction. For quadratic chi(2)-nonlinearity, this leads to variants of the Korteweg-de Vries equation (well known in fluid dynamics) with dispersion determined by the material properties. For narrow banded spectra, a corresponding envelope equation of nonlinear Schrodinger (NLS)-type, with full dispersive properties, is derived. Special attention is given to translate the NLS-solution to the physical field, which involves phase adaptations that contribute to the nonlinear dispersion relation. Then the propagation and distortion of double pumped pulses is studied by deriving uniformly valid analytic approximations. It is found, confirming but specifying previous observations, that when the quotient of amplitude and frequency difference is not small, side bands from third order effects have a contribution comparable to that of the first order terms. The uni-directional model describes the asymmetry in the distortions that are not described by the standard NLS-equation but which can be recovered when higher order dispersive effects are incorporated. The final conclusion when comparing the different models is that the uni-directional model is preferred above the NLS-model, based on its more general applicability for broad signals, its direct description of the physical fields, and the more direct analytical methods to find asymptotically valid approximations. [References: 10]
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