Modulational instability and numerically persistent solitary waves in a fractional Gross–Pitaevskii equation with Gaussian nonlocal interactions

We study a one-dimensional fractional Gross–Pitaevskii equation with Gaussian nonlocal interactions and competing cubic–quintic nonlinearities as a reduced model for anomalous matter-wave transport. An explicit plane-wave modulational-instability law is derived in which the fractional kinetic penalty, the Gaussian spectral filter, and the quintic saturation threshold appear as separate controls. The same symbol-resolved quantities are connected to localized profiles through peak, width, and cutoff-scale diagnostics. Localized waves are computed by a fractional-symbol-preconditioned constrained iteration and assessed by a Hamiltonian–Krein/VK index checklist, convergence tests, conservation-error diagnostics, long-time propagation, multiple perturbation amplitudes, and two- and three-parameter persistence maps. The results show that decreasing the fractional order broadens the MI window and produces taller, narrower pulses; increasing the quintic coefficient limits high-density compression; and increasing the interaction length smooths and broadens the localized state. The evidence supports finite-time numerical persistence in the tested regimes while avoiding a claim of unconditional orbital stability.