  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.
