Cooper, F.,Khare, A.,Comech, A.,Mihaila, B.,Dawson, J. F.,Saxena, A.
We discuss the stability properties of the solutions of the general nonlinear Schroedinger equation (NLSE) in 1+1 dimensions in an external potential derivable from a parity-time (PT) symmetric superpotential $W(x)$ that we considered earlier [Kevrekedis et al Phys. Rev. E 92, 042901 (2015)]. In particular we consider the nonlinear partial differential equation $\{ i \partial_t + \partial_x^2 - V^{-}(x) +| \psi(x,t) |^{2\kappa} \} \, \psi(x,t) = 0$, for arbitrary nonlinearity parameter $\kappa$. We study the bound state solutions when $V^{-}(x) = (1/4- b^2)$ sech$^2(x)$, which can be derived from two different superpotentials $W(x)$, one of which is complex and $PT$ symmetric. Using Derrick's theorem, as well as a time dependent variational approximation, we derive exact analytic results for the domain of stability of the trapped solution as a function of the depth $b^2$ of the external potential. We compare the regime of stability found from these analytic approaches with a numerical linear stability analysis using a variant of the Vakhitov-Kolokolov (V-K) stability criterion. The numerical results of applying the V-K condition give the same answer for the domain of stability as the analytic result obtained from applying Derrick's theorem. Our main result is that for $\kappa>2$ a new regime of stability for the exact solutions appears as long as $b > b_{crit}$, where $b_{crit}$ is a function of the nonlinearity parameter $\kappa$. In the absence of the potential the related solitary wave solutions of the NLSE are unstable for $\kappa>2$.