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The generalized nonlinear Schr\"odinger equation with full dispersion (FDNLS)is considered in the semiclassical regime. The Whitham modulation equations areobtained for the FDNLS equation with general linear dispersion and ageneralized, local nonlinearity. Assuming the existence of a four-parameterfamily of two-phase solutions, a multiple-scales approach yields a system offour independent, first order, quasi-linear conservation laws of hydrodynamictype that correspond to the slow evolution of the two wavenumbers, mass, andmomentum of modulated periodic traveling waves. The modulation equations arefurther analyzed in the dispersionless and weakly nonlinear regimes. Theill-posedness of the dispersionless equations corresponds to the classicalcriterion for modulational instability (MI). For modulations of linear waves,ill-posedness coincides with the generalized MI criterion, recently identifiedby Amiranashvili and Tobisch (New J. Phys. 21 (2019)). A new instability indexis identified by the transition from real to complex characteristics for theweakly nonlinear modulation equations. This instability is associated withlong-wavelength modulations of nonlinear two-phase wavetrains and can existeven when the corresponding one-phase wavetrain is stable according to thegeneralized MI criterion. Another interpretation is that, while infinitesimalperturbations of a periodic wave may not grow, small but finite amplitudeperturbations may grow, hence this index identifies a nonlinear instabilitymechanism for one-phase waves. Classifications of instability indices formultiple FDNLS equations with higher order dispersion, including applicationsto finite depth water waves and the discrete NLS equation are presented andcompared with direct numerical simulations.

Whitham modulation theory and two-phase instabilities for generalized nonlinear Schr\"{o}dinger equations with full dispersion