Nonlinear localized modes in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi mathvariant="script">PT</mml:mi></mml:math>-symmetric Rosen-Morse potential wells

Abstract

We report the existence and properties of localized modes described by a nonlinear Schr\"odinger equation with a complex $\mathcal{PT}$-symmetric Rosen-Morse potential well. Exact analytical expressions of the localized modes are found in both one-dimensional and two-dimensional geometry with self-focusing and self-defocusing Kerr nonlinearity. Linear stability analysis reveals that these localized modes are unstable for all real values of the potential parameters, although the corresponding linear Schr\"odinger eigenvalue problem possesses unbroken $\mathcal{PT}$ symmetry. This result has been verified by the direct numerical simulation of the governing equation. The transverse power-flow density associated with these localized modes has also been examined.