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Competing nonlinearities, such as the cubic (Kerr) and quintic nonlinearterms whose strengths are of opposite signs (the coefficients in front of thenonlinearities), exist in various physical media (in particular, in optical andmatter-wave media). A benign competition between self-focusing cubic andself-defocusing quintic nonlinear nonlinearities (known as cubic-quintic model)plays an important role in creating and stabilizing the self-trapping ofD-dimensional localized structures, in the contexts of standard nonlinearSchr\"{o}dinger equation. We incorporate an external periodic potential (linearlattice) into this model and extend it to the space-fractional scenario thatbegins to surface in very recent years---the nonlinear fractionalSchr\"{o}dinger equation (NLFSE), therefore obtaining the cubic-quintic or thepurely quintic NLFSE, and investigate the propagation and stability propertiesof self-trapped modes therein. Two types of one-dimensional localized gap modesare found, including the fundamental and dipole-mode gap solitons. Employingthe techniques based on the linear-stability analysis and direct numericalsimulations, we get the stability regions of all the localized modes; andparticularly, the anti-Vakhitov-Kolokolov criterion applies for the stableportions of soliton families generated in the frameworks of quintic-onlynonlinearity and competing cubic-quintic nonlinear terms.

nonlinear dynamics

One-dimensional gap solitons in quintic and cubic-quintic Fractional  nonlinear Schr\"{o}dinger equations with a periodically modulated linear  potential