  We discuss the effect of small perturbation on nodeless solutions of the
nonlinear \Schrodinger\ equation in 1+1 dimensions in an external complex
potential derivable from a parity-time symmetric superpotential that was
considered earlier [Phys.~Rev.~E 92, 042901 (2015)]. In particular we consider
the nonlinear partial differential equation $\{ \, \rmi \, \partial_t +
\partial_x^2 + g |\psi(x,t)|^2 - V^{+}(x) \, \} \, \psi(x,t) = 0$, where
$V^{+}(x) = \qty( -b^2 - m^2 + 1/4 ) \, \sech^2(x) - 2 i \, m \, b \, \sech(x)
\, \tanh(x)$ represents the complex potential. Here we study the perturbations
as a function of $b$ and $m$ using a variational approximation based on a
dissipation functional formalism. We compare the result of this variational
approach with direct numerical simulation of the equations. We find that the
variational approximation works quite well at small and moderate values of the
parameter $b m$ which controls the strength of the imaginary part of the
potential. We also show that the dissipation functional formalism is equivalent
to the generalized traveling wave method for this type of dissipation.
