Generalized anti-cubic nonlinear wave instability under spatiotemporal dispersion

This letter investigates modulational instability (MI) in an extended nonlinear Schrödinger equation incorporating generalized anti-cubic nonlinearities and spatiotemporal dispersion (STD). Linear stability analysis yields explicit instability criteria, gain spectra, and instability domains, revealing how inverse-power nonlinear responses reshape MI thresholds and amplification bands. The generalized nonlinearity index and the background amplitude are identified as key control parameters governing instability bandwidths, dominant perturbation modes, and nonlinear localization. Numerical simulations show that, in the absence of STD, MI leads to symmetric sideband growth, wave breakup, and saturation into localized structures. When STD is included, the instability spectrum is significantly modified, producing asymmetric amplification and richer nonlinear dynamics. The sign of the STD coefficient emerges as a geometric control parameter governing the orientation, coherence, and fragmentation of MI-induced structures. A direct connection with self-steepening effects is established, highlighting potential applications to ultrashort-pulse propagation in optical fibers and related nonlinear dispersive media.