Dynamical behaviors of optical solitons in parity–time (PT) symmetric sextic anharmonic double-well potentials

doi:10.1016/j.physleta.2015.06.034

Physics Letters A

Volume 379, Issue 36, 25 September 2015, Pages 2025-2029

https://doi.org/10.1016/j.physleta.2015.06.034 Get rights and content

Highlights

•
The NLS equation with $PT$ -symmetric anharmonic double-well potentials is studied.
•
Parameter regions are presented for the unbroken/broken $PT$ symmetry.
•
The families of numerical nonlinear modes and exact bright solitons are found.
•
The parameter regions are presented for stable and unstable solitons.

Abstract

We investigate optical solitons of the self-focusing nonlinear Schrödinger equation with novel $PT$ -symmetric sextic anharmonic double-well potentials. The parameter regions are presented for the $PT$ -symmetric Hamiltonian having unbroken/broken $PT$ symmetry. We examine the dynamical stability of the first several families of nonlinear modes in unbroken/broken $PT$ symmetric regions via numerical simulations.

Introduction

Dirac Hermiticity is a sufficient not necessary condition for a Hamiltonian to show entirely real spectra in the theory of quantum mechanics [1]. In 1998, Bender and Boettcher first found that a class of non-Hermitian $PT$ -symmetric Hamiltonians $H = - \partial_{x}^{2} + x^{2} {(i x)}^{ϵ}$ could also exhibit entirely real spectra for $ϵ \geq 0$ [2], that is to say, $ϵ = 0$ is a critical threshold in determining whether a phase transition can occur in the class of $PT$ -symmetric Hamiltonian systems. Here the parity reflection operator $P$ and time-reversal operator $T$ are defined by $x \to - x$ and $i \to - i$ , respectively [1], [2], [3]. Up to now, many non-Hermitian Hamiltonians with $PT$ -symmetric potentials have been shown to admit entirely real spectra (see, e.g., [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]). Moreover, there exist some experimental observations of $PT$ -symmetric phase transition [13], [14], [15], [16], [17], [18]. Recently, some types of complex $PT$ -symmetric potentials have been introduced in the nonlinear Schrödinger equation and coupled systems in nonlinear optics such that stable solitons were found such as the periodic, hyperbolic, and harmonic potentials [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], in which $PT$ -symmetric optical potentials arise from the complex refractive index distribution $n (x) = n_{R} (x) + i n_{I} (x)$ with $n_{R} (x) = n_{R} (- x)$ and $n_{I} (- x) = - n_{I} (x)$ [13], [14], [15].

The linear Schrödinger equations with real anharmonic potentials $α x^{2} + β x^{2 m}$ ( $x, α, β \in R$ , $m \in Z^{+}$ ) were studied in [37], [38], [39], in which the anharmonic potentials can be regarded as the perturbed case of the harmonic potential. Moreover, the periodic potential of a magneto-optical trap can be approximated via anharmonic potentials. In this paper, we will study the nonlinear Schrödinger equation (NLS) with the $PT$ -symmetric potential, whose real and imaginary parts are the sextic anharmonic double-well potential [38] and Hermite-super-Gaussian gain-or-loss distribution, respectively. We present parameter regions for the $PT$ -symmetric Hamiltonian having the unbroken/broken $PT$ symmetry. We give the families of numerical nonlinear modes and find its exact bright solitons with zero propagation constant, as well as examine their linear stability in unbroken/broken $PT$ -symmetric regions via numerical simulations. These results may be useful to design a corresponding optical $PT$ -symmetric experiment.

Section snippets

Nonlinear physical model with the $PT$ -symmetric potential and general theory

Here we consider the propagation of the laser beam along the z-axis of a medium with a transverse distribution of the refractive index and modulation of the gain-or-loss, and self-focusing Kerr nonlinearity, which can be described by the NLS equation with the $PT$ -symmetric potential [19], [30] $i \frac{\partial ψ}{\partial z} = - \frac{\partial^{2} ψ}{\partial x^{2}} + [V (x) + i W (x)] ψ - {| ψ |}^{2} ψ,$ where x and z are the transverse coordinate and propagation distance, respectively, $ψ (x, z)$ is an electric field, $V (x)$ is a real refractive index profile, $W (x)$ is a real

Linear stability of stationary solutions

Next we study the linear stability of solutions of Eq. (1) for some parameters. To show the linear stability of the nonlinear mode $ψ (x, z) = ϕ (x) e^{i μ z}$ , we considered a perturbed solution [40] $ψ (x, z) = ϕ (x) e^{i μ z} + ϵ [f (x) e^{i δ z} + g^{⁎} (x) e^{- i δ^{⁎} z}] e^{i μ z},$ where $ϵ ≪ 1$ , $f (x)$ and $g (x)$ are eigenfunctions of the linearized eigenvalue problem. Substituting (5) into Eq. (1) and linearizing with respect to ϵ we obtain the linear eigenvalue problem $(\begin{matrix} L & ϕ^{2} (x) \\ - ϕ^{⁎ 2} (x) & - L^{⁎} \end{matrix}) (\begin{matrix} f \\ g \end{matrix}) = δ (\begin{matrix} f \\ g \end{matrix}),$ where $L = \partial_{x}^{2} - V (x) - i W (x) + 2 {| ϕ (x) |}^{2} - μ$ . The linear

Conclusion

In conclusion, we have investigated numerical solution families and presented exact optical bright solitons of the nonlinear Schrödinger equation with novel $PT$ -symmetric anharmonic double-well potentials. We analyze the parameter regions for the Hamiltonian having unbroken/broken $PT$ symmetry. We show the linear stability of these bright solitons for some parameters using numerical simulations. For the fixed amplitude of gain-or-loss distribution, we find that the super-Gaussian frequency

Acknowledgements

The authors would like to thank the referees for their valuable comments and suggestions. The work was partially supported by NSFC (No. 61178091) and NKBRPC (2011CB302400).

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Dynamical behaviors of optical solitons in parity–time (PT) symmetric sextic anharmonic double-well potentials

Highlights

Abstract

Introduction

Section snippets

Nonlinear physical model with the PT-symmetric potential and general theory

Linear stability of stationary solutions

Conclusion

Acknowledgements

Phys. Lett. A

Ann. Phys. (NY)

Phys. Rep.

Making sense of non-Hermitian Hamiltonians

Rep. Prog. Phys.

Real spectra in non-Hermitian Hamiltonians having PT symmetry

Phys. Rev. Lett.

Families of particles with different masses in PT-symmetric quantum field theory

Phys. Rev. Lett.

Shape invariant potentials with PT symmetry

J. Phys. A

Theory of coupled optical PT-symmetric structures

Opt. Lett.

The PT-symmetric Rosen–Morse II potential: effects of the asymptotically non-vanishing imaginary potential component

J. Phys. A

Pseudo-Hermiticity versus PT-symmetry: the necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian

J. Math. Phys.

Beam dynamics in PT symmetric optical lattices

Phys. Rev. Lett.

Spectral singularities of complex scattering potentials and infinite reflection and transmission coefficients at real energies

Phys. Rev. Lett.

Naimark-dilated PT-symmetric brachistochrone

Phys. Rev. Lett.

PT-symmetric optical superlattices

J. Phys. A

Observation of PT-symmetry breaking in complex optical potentials

Phys. Rev. Lett.

Exponentially fragile PT symmetry in lattices with localized eigenmodes

Phys. Rev. Lett.

Observation of parity–time symmetry in optics

Nat. Phys.

Parity–time synthetic photonic lattices

Nature (London)

Supersymmetric optical structures

Phys. Rev. Lett.

Parity–time-symmetric microring lasers

Science

Optical solitons in PT periodic potentials

Phys. Rev. Lett.

Stability of solitons in parity–time-symmetric couplers

Opt. Lett.

Dynamical behaviors of optical solitons in parity–time ( $PT$ ) symmetric sextic anharmonic double-well potentials

Nonlinear physical model with the $PT$ -symmetric potential and general theory