We address the existence and stability of spatial localized modes supported by a parity-time-symmetric complex potential in the presence of power-law nonlinearity. The analytical expressions of the localized modes, which are Gaussian in nature, are obtained in both (1+1) and (2+1) dimensions. A linear stability analysis corroborated by the direct numerical simulations reveals that these analytical localized modes can propagate stably for a wide range of the potential parameters and for various order nonlinearities. Some dynamical characteristics of these solutions, such as the power and the transverse power-flow density, are also examined.

http://arxiv.org/abs/1404.7322
Quantum Physics (quant-ph); Pattern Formation and Solitons (nlin.PS); Exactly Solvable and Integrable Systems (nlin.SI)

The existence and stability of the nonlinear spatial localized modes are investigated in parity-time symmetric optical media characterized by a generic complex hyperbolic refractive index distribution with competing gain and loss profile. The exact analytical expressions of the localized modes are found for all values of the competing parameter and in the presence of both the self-focusing and self-defocusing Kerr nonlinearity. The effect of competing gain/loss profile on the stability structure of these localized modes are discussed with the help of linear stability analysis followed by the direct numerical simulation of the governing equation. The spatial localized modes in two-dimensional geometry as well as the transverse power-flow density associated with these localized modes are also examined.

http://arxiv.org/abs/1306.5983

Optics (physics.optics)

We report the existence and properties of localized modes described by nonlinear Schroedinger equation with complex 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 corresponding linear Schroedinger eigenvalue problem possesses unbroken 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.

http://arxiv.org/abs/1304.2105
Quantum Physics (quant-ph); Pattern Formation and Solitons (nlin.PS); Optics (physics.optics)

Using an appropriate change of variable, the Schroedinger equation is transformed into a second-order differential equation satisfied by recently discovered Jacobi type \(X_m\) exceptional orthogonal polynomials. This facilitates the derivation of infinite families of exactly solvable Hermitian as well as non-Hermitian trigonometric and hyperbolic Scarf potentials whose bound state solutions are associated with the aforesaid exceptional orthogonal polynomials. These infinite families of potentials are shown to be extensions of the conventional trigonometric and hyperbolic Scarf potentials by the addition of some rational terms characterized by the presence of classical Jacobi polynomials. All the members of a particular family of these `rationally extended polynomial-dependent’ potentials have the same energy spectrum and possess translational shape invariant symmetry. The obtained non-Hermitian potentials are shown to be quasi-Hermitian in nature ensuring the reality of the associated energy spectra.

http://arxiv.org/abs/1210.0119
Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Using the method of point canonical transformation, we derive some exactly solvable rationally extended quantum Hamiltonians which are non-Hermitian in nature and whose bound state wave functions are associated with Laguerre and Jacobi-type \(X_1\) exceptional orthogonal polynomials. These Hamiltonians are shown, with the help of imaginary shift of co-ordinate: \(e^{-\alpha p} x e^{\alpha p} = x+ i \alpha\), to be both quasi and pseudo-Hermitian. It turns out that the corresponding energy spectra is entirely real.

http://arxiv.org/abs/1205.5860
Mathematical Physics (math-ph); Quantum Physics (quant-ph)