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 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, Kevrekidis et al (2015 Phys. Rev. E 92 042901). In particular we consider the nonlinear partial differential equation {i partial derivative(t) + partial derivative(2)(x) - V-(x) + |psi(x, t)|(2k)}psi(x, t) = 0, for arbitrary nonlinearity parameter k. 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 k > 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 k. In the absence of the potential the related solitary wave solutions of the NLSE are unstable for k > 2.