We study solitons and vortices described by the (2+1)-dimensional fourth-order generalized nonlinear Schrödinger equation with cubic-quintic nonlinearity. Necessary conditions for the existence of such structures are investigated analytically using conservation laws and asymptotic behavior of localized solutions. We derive the generalized virial relation, which describes the combined influence of linear and nonlinear effects on the evolution of the wave packet envelope. By means of refined variational analysis, we predict the main features of steady soliton solutions, which have been shown to be in good agreement with our numerical results. Soliton and vortex stability is investigated by linear analysis and direct numerical simulations. We show that stable bright solitons exist in nonlinear Kerr media both in anomalous and normal dispersive regimes, even if only the fourth-order dispersive effect is taken into account. Vortices occur robust with respect to symmetry-breaking azimuthal instability only in the presence of additional defocusing quintic nonlinearity in the strongly nonlinear regime. We apply our results to the theoretical explanation of whistler self-induced waveguide propagation in plasmas, and discuss possible applications to light beam propagation in cubic-quintic optical materials and to solitons in two-dimensional molecular systems.

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The generalized nonlinear Schrödinger equation including second and fourth order dispersive effects, local and nonlocal nonlinearity has been used to find conditions for stable upper hybrid soliton formation. It is shown that for certain conditions this equation has two stable soliton solutions with equal number of quanta but with different spatial scales. Qualitative correspondence of the characteristic soliton features with some experimental data is shown.

The stability of two-dimensional bright vortex solitons in a media with focusing cubic and defocusing quintic nonlinearities is investigated analytically and numerically. It is proved that above some critical beam powers not only one-and two-charged but also multiple-charged stable vortex solitons do exist. A vortex soliton occurs robust with respect to symmetry-breaking modulational instability in the self-defocusing regime provided that its radial profile becomes flattened, so that a self-trapped wave beam gets a pronounced surface. It is demonstrated that the dynamics of a slightly perturbed stable vortex soliton resembles an oscillation of a liquid stream having a surface tension. Using the idea of sustaining effective surface tension for spatial vortex soliton in a media with competing nonlinearities the explanation of a suppression of the modulational instability is proposed.