We develop a model for the description of nonlinear pulse propagation in multimode optical fibers with a parabolic refractive index profile. It consists in a 1+1D generalized nonlinear Schrödinger equation with a periodic nonlinear coefficient, which can be solved in an extremely fast and efficient way. The model is able to quantitatively reproduce recently observed phenomena like geometric parametric instability and broadband dispersive wave emission. We envisage that our equation will represent a valuable tool for the study of spatiotemporal nonlinear dynamics in the growing field of multimode fiber optics.Nonlinear pulse propagation in multimode fibers (MMFs) is focusing a tremendous research interest [1] . Even if graded index (GRIN) MMFs have been invented long ago [2], it is only very recently that the systematic and in depth study of the complex nonlinear spatiotemporal effects that can take place in these fibers has began [3]. The experimental observations of multimode solitons [4,5], ultrabroadband dispersive waves [6,7], geometric parametric instability (GPI) [8][9][10], beam self-cleaning [11][12][13], and novel forms of supercontinuum [14,15] are striking examples of the incredibly rich and complex scenario offered by nonlinear propagation in GRIN fibers. The reason why these observations took so long to appear is that the study of spatiotemporal effect in MMFs is an intrinsically hard task from the experimental, theoretical and numerical point of view.The description of pulse propagation in MMFs must consider the three spatial and the temporal dimensions at the same time, because spatial and temporal effects cannot, in principle, be decoupled. Essentially two models are exploited for the mathematical description of propagation in MMFs: the 3+1D Generalized Nonlinear Schrödinger Equation (GNLSE) with a spatial potential, also named Gross-Pitaevskii Equation (GPE) in the context of Bose-Einsten condensates [4,5,8,17], and the multi-mode GNLSE (MM-GNLSE) [18]. In the GPE the transverse dimensions are accounted for in the propagation equation through the potential describing the refractive index profile of the fiber. In the MM-GNLSE the transverse dimensions are described indirectly, through the projection over the different fiber modes, which are coupled by the nonlinearity. GPE is the most direct tool, but also the most computationally expensive: for example, the simulation of the propagation of multimode solitons over a few meters of fiber requires several days of computation [4]. The computational complexity can be partially reduced by considering exclusively radially symmetric modes [19]. The MM-GNLSE, consisting in N coupled 1+1D GNLSE, permits to reduce the computational time only if a limited number of modes are excited (typically N < 10), because the number of nonlinear coupling terms grows as N 4 . When considering beams with a relative large size, taking into account only a few modes may lead to inaccurate results.In this Letter we develop a model for the description of nonlinear pulse propag...

We present a theoretical and numerical study of light propagation in graded-index (GRIN) multimode fibers where the core diameter has been periodically modulated along the propagation direction. The additional degree of freedom represented by the modulation permits to modify the intrinsic spatiotemporal dynamics which appears in multimode fibers. More precisely, we show that modulating the core diameter at a periodicity close to the self-imaging distance allows to induce a Moiré-like pattern, which modifies the geometric parametric instability gain observed in homogeneous GRIN fibers.