Bright solitons in defocusing media with spatial modulation of the quintic nonlinearity

TL;DR

This paper explores the existence and stability of bright solitons in one- and two-dimensional self-defocusing media with spatially modulated quintic nonlinearities, extending previous findings from cubic nonlinearities and providing analytical and numerical insights.

Contribution

It demonstrates the formation and stability of bright solitons in quintic self-defocusing media with spatial modulation, including exact solutions and a new variational ansatz.

Findings

Stable fundamental 1D solitons confirmed by eigenvalue analysis and simulations.

Higher-order 1D solitons with more than two nodes are unstable.

Exact 2D soliton solutions are constructed.

Abstract

It has been recently demonstrated that self-defocusing (SDF) media with the cubic nonlinearity, whose local coefficient grows from the center to periphery fast enough, support stable bright solitons, without the use of any linear potential. Our objective is to test the genericity of this mechanism for other nonlinearities, by applying it to one- and two-dimensional (1D and 2D) quintic SDF media. The models may be implemented in optics (in particular, in colloidal suspensions of nanoparticles), and the 1D model may be applied to the description of the Tonks-Girardeau gas of ultracold bosons. In 1D, the nonlinearity-modulation function is taken as $% g_{0}+\sinh ^{2}(\beta x) $. This model admits a subfamily of exact solutions for fundamental solitons. Generic \ soliton solutions are constructed in a numerical form, and also by means of the Thomas-Fermi and variational approximations (TFA and VA). In particular, a new ansatz for the VA is proposed, in the form of "raised $\mathrm{sech}$", which provides for an essentially better accuracy than the usual Gaussian ansatz. The stability of all the fundamental (nodeless) 1D solitons is established through the computation of the corresponding eigenvalues for small perturbations, and also verified by direct simulations. Higher-order 1D solitons with two nodes have a limited stability region, all the modes with more than two nodes being unstable. It is concluded that the recently proposed "anti-VK" stability criterion for fundamental bright solitons in systems with SDF nonlinearities holds here too. Particular exact solutions for 2D solitons are produced as well.