Abstract. Dispersion effects induce new instabilities and dynamics in the weakly nonlinear description of light propagation in fiber Bragg gratings. A new family of dispersive localized pulses that propagate with the group velocity is numerically found and its stability is also analyzed. The unavoidable different asymptotic order of transport and dispersion effects plays a crucial role in the determination of these localized states. These results are also interesting from the point of view of general pattern formation since this asymptotic imbalance is a generic situation in any transport dominated (i.e., nonzero group velocity) spatially extended system.1. Introduction. Fiber Bragg gratings (FBG) are microstructured optical fibers that present a spatially periodic variation of the refractive index. The combination of the guiding properties of the periodic media with the Kerr nonlinearity of the fiber results in the very particular light propagation characteristic of these elements, which make them very promising for many technological applications that range from optical communications (wavelength division, dispersion management, optical buffers and storing devices, etc.) to fiber sensing (structural stress measure in aircraft components and buildings, temperature change detection, etc.), see, e.g., the recent review [8].The amplitude equations that are commonly used in the literature to model one dimensional light propagation in a FBG are the so-called nonlinear coupled mode equations (NLCME) [17,6,5,1,7], which, conveniently scaled,can be written as

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