  The existence and stability of stable bright solitons in one-dimensional (1D)
media with a spatially periodical modulated Kerr nonlinearity are demonstrated
by means of the linear-stability analysis and in direct numerical simulations.
The nonlinear potential landscape can balance the fractional-order diffraction
and thus stabilizes the solitons, making the model unique and governed by the
recently introduced fractional Schr\"{o}dinger equation with a self-focusing
cubic nonlinear lattice. Both 1D fundamental and multihump solitons (in forms
of dipole and tripole ones) are found, which occupy one or three cells of the
nonlinear lattice respectively, depending on the soliton's power (intensity).
We find that the profiles of the predicted soliton families are impacted
intensely by the L\'{e}vy index $\alpha$ which denotes the level of fractional
Laplacian, so does to their stability. The stabilization of soliton families is
possible if $\alpha$ exceeds a threshold value, below which the balance between
fractional-order diffraction and the spatially modulated focusing nonlinearity
will be broken.
