  We discuss the stability properties of the solutions of the general nonlinear
\Schrodinger\ equation (NLSE) in 1+1 dimensions in an external potential
derivable from a parity-time ($\PT$) symmetric superpotential $W(x)$ that we
considered earlier \cite{PhysRevE.92.042901}. In particular we consider the
nonlinear partial differential equation $ \{
  i \,
  \partial_t
  +
  \partial_x^2
  -
  V(x)
  + g
  | \psi(x,t) |^{2\kappa}
  \} \, \psi(x,t)
  =
  0 \>, $ for arbitrary nonlinearity parameter $\kappa$, where $g= \pm1$ and
$V$ is the well known P{\"o}schl-Teller potential which we allow to be
repulsive as well as attractive. Using energy landscape methods, linear
stability analysis as well as a time dependent variational approximation, we
derive consistent analytic results for the domains of instability of these new
exact solutions as a function of the strength of the external potential and
$\kappa$. For the repulsive potential (and $g=+1$) we show that there is a
translational instability which can be understood in terms of the energy
landscape as a function of a stretching parameter and a translation parameter
being a saddle near the exact solution. In this case, numerical simulations
show that if we start with the exact solution, the initial wave function breaks
into two pieces traveling in opposite directions. If we explore the slightly
perturbed solution situations, a 1\% change in initial conditions can change
significantly the details of how the wave function breaks into two separate
pieces. For the attractive potential (and $g=+1$), changing the initial
conditions by 1 \% modifies the domain of stability only slightly. For the case
of the attractive potential and negative $g$ perturbed solutions merely
oscillate with the oscillation frequencies predicted by the variational
approximation.
