Self-trapped spatially localized states in combined linear-nonlinear periodic potentials

Abstract

We analyze the existence and stability of two kinds of self-trapped spatially localized gap modes, gap solitons and truncated nonlinear Bloch waves, in one-and two-dimensional optical or matter-wave media with selffocusing nonlinearity, supported by a combination of linear and nonlinear periodic lattice potentials. The former is found to be stable once placed inside a single well of the nonlinear lattice, it is unstable otherwise. Contrary to the case with constant self-focusing nonlinearity, where the latter solution is always unstable, here, we demonstrate that it nevertheless can be stabilized by the nonlinear lattice since the model under consideration combines the unique properties of both the linear and nonlinear lattices. The practical possibilities for experimental realization of the predicted solutions are also discussed.

Preventing critical collapse of higher-order solitons by tailoring unconventional optical diffraction and nonlinearities

“…6a, d, b, e, respectively, and as unstable object when its propagation constant approaches the edge of the band gap in Fig. 6c, f. We stress that the multiple gap solitons or gap soliton clusters reported here are equivalent to the recently found truncated nonlinear Bloch waves or gap waves in nonlinear optical media and ultracold atoms 17,23,[64][65][66][67][68] .…”

“…An obvious conceptional extension of our study is to reveal the underlying physical mechanism for critical collapse related to physical parameters including Lévy index α (diffraction order) and order of a single nonlinearity (e.g., cubic only) as well as the strength of optical lattices. Rich localized modes and nonlinear dynamics are yet to be explored in an incommensurate structure -combined linear and nonlinear lattices [21][22][23] . Another interesting issue is to study the localization and delocalization properties of waves in other periodic lattices 69 with regolabile (fractional) diffraction.…”

“…Although nonlinear lattices with smooth variation of nonlinearity may support various species of solitons, to stabilize multidimensional solitons using sinusoidal nonlinear lattices remains an open area of research 7 . Further, the combined linear-nonlinear lattices yield stabilization of various soliton families [21][22][23] .…”

One-dimensional gap solitons in quintic and cubic–quintic fractional nonlinear Schrödinger equations with a periodically modulated linear potential

“…Actually, finding stable fundamental gap solitons in quintic self-focusing nonlinearity is still an open issue, to our knowledge. Instead of being unstable objects, such localized gap modes can be stable solutions pinned at the first and second finite band gaps (of the underlying linear spectrum) under the cubic self-focusing nonlinearity [74]. Dynamic propagation properties of the gap solitons against arbitrarily small initial perturbations, supported by the NLFSE model with an optical lattice embedded with quintic nonlinearity, are portrayed in Fig.…”

“…Particularly, the competing cubic-quintic nonlinear lattices (both the selffocusing cubic and self-defocusing quintic nonlinear terms are integrated with nonlinear lattices, with commensurate and incommensurate periods of the two lattices) were predicted to be a feasible and effective way to generate stable 2D solitons and vortices [65]. The solitons in the models with combined linear and nonlinear lattices have been and are still being comprehensively researched in recent years [71][72][73][74]. The purely nonlinear defocusing media with spatially inhomogeneous nonlinearity whose local strength grows quickly enough from the center toward periphery in the D-dimensional coordinate, which are built on self-defocusing background and therefore do not possess critical and supercritical collapsestypical characteristics for solitons in self-focusing media, enriched the generation of various families of stable solitons and soliton composites in the self-trapping regime [75][76][77][78][79][80][81][82][83][84][85][86][87][88], such as the fundamental solitons for all (D-dimensional) space coordiantes [75,76], 1D multihump states in forms of dipole and multipole solitons [75][76][77], 2D bright solitary vortices carrying with an arbitrarily high topological charge [75,76], 2D localized dark solitons and vortices [86], multifarious 3D localized modes that are comprised of soliton gyroscopes [81] and skyrmions [82], and very recently the flat-top solitons (in both 1D and 2D spaces) and 2D vortices [88,89], to name just some of them.…”