  We consider a recently proposed nonlinear Schroedinger equation exhibiting
soliton-like solutions of the power-law form $e_q^{i(kx-wt)}$, involving the
$q$-exponential function which naturally emerges within nonextensive
thermostatistics [$e_q^z \equiv [1+(1-q)z]^{1/(1-q)}$, with $e_1^z=e^z$]. Since
these basic solutions behave like free particles, obeying $p=\hbar k$, $E=\hbar
\omega$ and $E=p^2/2m$ ($1 \le q<2$), it is relevant to investigate how they
change under the effect of uniform acceleration, thus providing the first steps
towards the application of the aforementioned nonlinear equation to the study
of physical scenarios beyond free particle dynamics. We investigate first the
behaviour of the power-law solutions under Galilean transformation and discuss
the ensuing Doppler-like effects. We consider then constant acceleration,
obtaining new solutions that can be equivalently regarded as describing a free
particle viewed from an uniformly accelerated reference frame (with
acceleration $a$) or a particle moving under a constant force $-ma$. The latter
interpretation naturally leads to the evolution equation $i\hbar
\frac{\partial}{\partial t}(\frac{\Phi}{\Phi_0}) = -
\frac{1}{2-q}\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2}
[(\frac{\Phi}{\Phi_0})^{2-q}] + V(x)(\frac{\Phi}{\Phi_0})^{q}$ with $V(x)=max$.
Remarkably enough, the potential $V$ couples to $\Phi^q$, instead of coupling
to $\Phi$, as happens in the familiar linear case ($q=1$).
