Stabilization of 1D solitons by fractional derivatives in systems with quintic nonlinearity

We study theoretically the properties of a soliton solution of the fractional Schrödinger equation with quintic nonlinearity. Under “fractional” we understand the Schrödinger equation, where ordinary Laplacian (second spatial derivative in 1D) is substituted by its fractional counterpart with Lévy index $$\alpha$$ α . We speculate that the latter substitution corresponds to phenomenological account for disorder in a system. Using analytical (variational and perturbative) and numerical arguments, we have shown that while in the case of Schrödinger equation with the ordinary Laplacian (corresponding to Lévy index $$\alpha =2$$ α = 2 ), the soliton is unstable, even infinitesimal difference $$\alpha$$ α from 2 immediately stabilizes the soliton texture. Our analytical and numerical investigations of $$\omega (N)$$ ω ( N ) dependence ($$\omega$$ ω is soliton frequency and N its mass) show (within the famous Vakhitov–Kolokolov criterion) the stability of our soliton texture in the fractional $$\alpha <2$$ α < 2 case. Direct numerical analysis of the linear stability problem of soliton texture also confirms this point. We show analytically and numerically that fractional Schrödinger equation with quintic nonlinearity admits the existence of (stable) soliton textures at $$2/3<\alpha <2$$ 2 / 3 < α < 2 , which is in accord with existing literature data. These results may be relevant to both Bose–Einstein condensates in cold atomic gases and optical solitons in the disordered media.

Stable high dimensional solitons in nonlocal competing cubic-quintic nonlinear media

We find and stabilize high dimensional dipole and quadrupole solitons in nonlocal competing cubic-quintic nonlinear media. By adjusting the propagation constant, cubic and quintic nonlinear coefficients, the stable intervals for dipole and quadrupole solitons which are parallel to $x$ axis and ones after rotating 45 degrees counterclockwise around the origin of coordinate are found. For the dipole solitons and ones after rotating, their stability is controlled by the propagation constant, the coefficients of cubic and quintic nonlinearity. For the quadrupole solitons, their stability is controlled by the propagation constant and the coefficient of cubic nonlinearity, rather than the coefficient of quintic nonlinearity, though there is a small effect of the quintic nonlinear coefficient on the stability. Our proposal may provide a way to generate and stabilize some novel high dimensional nonlinear modes in nonlocal system.

Stabilization of 1D Solitons By Fractional Derivatives In Systems With Quintic Nonlinearity

We study theoretically the properties of a soliton solution of the fractional Schrödinger equation with quintic nonlinearity. Under ”fractional” we understand the Schrödinger equation, where ordinary Laplacian (second spatial derivative in 1D) is substituted by its fractional counterpart with Lévy index α. We speculate that the latter substitution corresponds to phenomenological account for disorder in a system. Using analytical (variational and perturbative) and numerical arguments, we have shown that while in the case of Schrödinger equation with the ordinary Lapla-cian (corresponding to Lévy index α = 2), the soliton is unstable, even infinitesimal difference α from 2 immediately stabilizes the soliton texture. Our analytical and numerical investigations of ω(N) dependence (ω is soliton frequency and N its mass) show (within the famous Vakhitov-Kolokolov criterion) the stability of our soliton texture in the fractional α < 2 case. Direct numerical analysis of the linear stability problem of soliton texture also confirms this point. We show analytically and numerically that fractional Schrödinger equation with quintic nonlinearity admits the existence of (stable) soliton textures at 2/3 < α < 2, which is in accord with existing literature data. These results may be relevant to both Bose-Einstein condensates in cold atomic gases and optical solitons in the disordered media.

Dissipative structures in the resonant interaction of laser radiation with nonlinear dispersive medium

The article presents the results of studies on the stability of dissipative structures (DS) arising in the resonant interaction of laser radiation with a nonlinear medium. Resonant interaction is modeled by the one dimensional complex Ginzburg-Landau equation with a nonconservative cubic–quintic nonlinearity. The areas of existence of stable DS solutions have been determined analytically using a variational approach and confirmed numerically by extensive numerical simulations.

Quiescent Gap Solitons in Coupled Nonuniform Bragg Gratings with Cubic-Quintic Nonlinearity

We study the stability characteristics of zero-velocity gap solitons in dual-core Bragg gratings with cubic-quintic nonlinearity and dispersive reflectivity. The model supports two disjointed families of gap solitons (Type 1 and Type 2). Additionally, asymmetric and symmetric solitons exist in both Type 1 and Type 2 families. A comprehensive numerical stability analysis is performed to analyze the stability of solitons. It is found that dispersive reflectivity improves the stability of both types of solitons. Nontrivial stability boundaries have been identified within the bandgap for each family of solitons. The effects and interplay of dispersive reflectivity and the coupling coefficient on the stability regions are also analyzed.