Dynamical behaviors of optical solitons in parity–time () symmetric sextic anharmonic double-well potentials
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 -symmetric Hamiltonians could also exhibit entirely real spectra for [2], that is to say, is a critical threshold in determining whether a phase transition can occur in the class of -symmetric Hamiltonian systems. Here the parity reflection operator and time-reversal operator are defined by and , respectively [1], [2], [3]. Up to now, many non-Hermitian Hamiltonians with -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 -symmetric phase transition [13], [14], [15], [16], [17], [18]. Recently, some types of complex -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 -symmetric optical potentials arise from the complex refractive index distribution with and [13], [14], [15].
The linear Schrödinger equations with real anharmonic potentials (, ) 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 -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 -symmetric Hamiltonian having the unbroken/broken 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 -symmetric regions via numerical simulations. These results may be useful to design a corresponding optical -symmetric experiment.
Section snippets
Nonlinear physical model with the -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 -symmetric potential [19], [30] where x and z are the transverse coordinate and propagation distance, respectively, is an electric field, is a real refractive index profile, 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 , we considered a perturbed solution [40] where , and are eigenfunctions of the linearized eigenvalue problem. Substituting (5) into Eq. (1) and linearizing with respect to ϵ we obtain the linear eigenvalue problem where . The linear
Conclusion
In conclusion, we have investigated numerical solution families and presented exact optical bright solitons of the nonlinear Schrödinger equation with novel -symmetric anharmonic double-well potentials. We analyze the parameter regions for the Hamiltonian having unbroken/broken 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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