Stability of exact solutions of the nonlinear Schroedinger equation in an external potential having supersymmetry and parity-time symmetry

TL;DR

This paper analyzes the stability of exact solutions to the nonlinear Schrödinger equation with PT-symmetric potentials, deriving analytic stability domains and confirming them with numerical methods, revealing new stability regimes for certain nonlinearities.

Contribution

It provides the first analytic derivation of stability domains for NLSE solutions with PT-symmetric potentials, validated by numerical analysis, and identifies new stability regimes for nonlinearities greater than 2.

Findings

Exact stability domains derived analytically using Derrick's theorem.

Numerical Vakhitov-Kolokolov criterion confirms analytic stability results.

New stability regime identified for nonlinear parameter κ > 2 when potential depth exceeds a critical value.

Abstract

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 $\kappa>2$.