  Extending the recent work on models with spatially nonuniform nonlinearities,
we study bright solitons generated by the nonpolynomial self-defocusing (SDF)
nonlinearity in the framework of the one-dimensional (1D) Mu\~{n}oz-Mateo -
Delgado (MM-D) equation (the 1D reduction of the Gross-Pitaevskii equation with
the SDF nonlinearity), with the local strength of the nonlinearity growing at
any rate faster than |x| at large values of coordinate x. We produce numerical
solutions and analytical ones, obtained by means of the Thomas-Fermi
approximation (TFA), for nodeless ground states, and for excited modes with 1,
2, 3, and 4 nodes, in two versions of the model, with the steep (exponential)
and mild (algebraic) nonlinear-modulation profiles. In both cases, the ground
states and the single-node ones are completely stable, while the stability of
the higher-order modes depends on their norm (in the case of the algebraic
modulation, they are fully unstable). Unstable states spontaneously evolve into
their stable lower-order counterparts.
