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We discuss the effect of small perturbation on nodeless solutions of thenonlinear \Schrodinger\ equation in 1+1 dimensions in an external complexpotential derivable from a parity-time symmetric superpotential that wasconsidered earlier [Phys.~Rev.~E 92, 042901 (2015)]. In particular we considerthe 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 perturbationsas a function of $b$ and $m$ using a variational approximation based on adissipation functional formalism. We compare the result of this variationalapproach with direct numerical simulation of the equations. We find that thevariational approximation works quite well at small and moderate values of theparameter $b m$ which controls the strength of the imaginary part of thepotential. We also show that the dissipation functional formalism is equivalentto the generalized traveling wave method for this type of dissipation.

journal of physics. a, mathematical and theoretical

Response of exact solutions of the nonlinear Schrodinger equation to  small perturbations in a class of complex external potentials having  supersymmetry and parity-time symmetry