Abstract
Self-trapped modes suffer critical collapse in two-dimensional cubic systems. To overcome such a collapse, linear periodic potentials or competing nonlinearities between self-focusing cubic and self-defocusing quintic nonlinear terms are often introduced. Here, we combine both schemes in the context of an unconventional and nonlinear fractional Schrödinger equation with attractive-repulsive cubic–quintic nonlinearity and an optical lattice. We report theoretical results for various two-dimensional trapped solitons, including fundamental gap and vortical solitons as well as the gap-type soliton clusters. The latter soliton family resembles the recently-found gap waves. We uncover that, unlike the conventional case, the fractional model exhibiting fractional diffraction order strongly influences the formation of higher band gaps. Hence, a new route for the study of self-trapped modes in these newly emergent higher band gaps is suggested. Regimes of stability and instability of all the soliton families are obtained with the help of linear-stability analysis and direct simulations.
Highlights
Self-trapped modes suffer critical collapse in two-dimensional cubic systems
We propose and demonstrate, theoretically and numerically, a framework of 2D nonlinear fractional Schrödinger equation (NLFSE)—describing light propagation in a nonlinear periodic system with an optical lattice and attractive-repulsive cubic–quintic nonlinearities—which can suppress the critical collapse mentioned before
We start from the cubic–quintic NLFSE, which describes a laser beam propagation in a nonlinear medium and can be expressed in scaled form: i
Summary
Self-trapped modes suffer critical collapse in two-dimensional cubic systems. To overcome such a collapse, linear periodic potentials or competing nonlinearities between self-focusing cubic and self-defocusing quintic nonlinear terms are often introduced. The periodic potentials play a paramount stabilizing part for D-dimensional localized modes including fundamental solitons and more complicated states like gap solitons, multihump states, and vortex solitons[4,7,8]. Another potential scheme is assisted by nonlinear pseudopotentials—the well-known nonlinear lattices[7,18,19,20]. Materials which exhibit different orders of nonlinearity, including the well-known cubic–quintic model with competing nonlinearities (self-focusing cubic and self-defocusing quintic nonlinear terms), can create stable multidimensional solitons and suppress the above-mentioned critical- and supercritical collapses[4,7,8,21,22]. Thevery recently confirmed the existence of stable multidimensional states of ultracold bosonic gases[8], appearing as the so-called quantum droplets[27,28], where the Bose–Bose mixture collapses are suppressed by quantum fluctuations[33]
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