Modulational instability in the nonlocal \(\chi ^{(2)}\)-model
The paper deals with a model approximating the nonlinear propagation of waves in \(\chi^{(2)}\) media. The proposed model is a nonlocal version of the Nonlinear Schrödinger Equation and resembles the generic Kerr type nonlocal model but with different nonlocal term that retains some phase information. The main goal of the paper is to link the dynamical properties of the proposed model with those of the full \(\chi^{(2)}\)-model, with the emphasis on the modulational instability (MI) properties. It is shown that the MI is of the oscillatory type and of finite bandwidth. Moreover, it is possible to identify regions in the parameter space for which a fundamental gain band exists, and regions for which higher order gain bands and modulational stability exist. It is shown that the MI analysis for the nonlocal model is applicable in the finite walk-off rank and that the plane waves of the nonlocal \(\chi^{(2)}\)-model are recovered as the asymptotic limit of one of the branches of the plane waves of the full \(\chi^{(2)}\)-model by means of a singular perturbation approach.
scientific article; zbMATH DE number 5142354