PT-symmetric Potentials Research Articles

Darboux transformation and explicit solutions for the [formula omitted]-dimensional nonlocal nonlinear Schrödinger–Maxwell–Bloch system

The (2+1)-dimensional nonlocal nonlinear Schrödinger–Maxwell–Bloch system with PT-symmetric potentials will be investigated in this paper. The (2+1)-dimensional Darboux transformation will be constructed, some explicit solutions including periodic waves solutions, soliton solutions and complexiton solutions will be obtained under different backgrounds, and the interaction behaviors of those solutions will be discussed through some figures.

Parity-time symmetry in coherent asymmetric double quantum wells

A coherently prepared asymmetric double semiconductor quantum well (QW) is proposed to realize parity-time (PT) symmetry. By appropriately tuning the laser fields and the pertinent QW parameters, PT-symmetric optical potentials are obtained by three different methods. Such a coherent QW system is reconfigurable and controllable, and it can generate new approaches of theoretically and experimentally studying PT-symmetric phenomena.

Bifurcation of nonlinear bound states in the periodic Gross-Pitaevskii equation with 𝒫𝒯-symmetry

AbstractThe stationary Gross–Pitaevskii equation in one dimension is considered with a complex periodic potential satisfying the conditions of the 𝒫𝒯 (parity-time reversal) symmetry. Under rather general assumptions on the potentials, we prove bifurcations of 𝒫𝒯-symmetric nonlinear bound states from the end points of a real interval in the spectrum of the non-selfadjoint linear Schrödinger operator with a complex 𝒫𝒯-symmetric periodic potential. The nonlinear bound states are approximated by the effective amplitude equation, which bears the form of the cubic nonlinear Schrödinger equation. In addition, we provide sufficient conditions for the appearance of complex spectral bands when the complex 𝒫𝒯-symmetric potential has an asymptotically small imaginary part.

Inverse scattering solutions and dynamics for a nonlocal nonlinear Gross–Pitaevskii equation with PT-symmetric external potentials

A generalized nonlocal nonlinear Gross–Pitaevskii(GP) equation is presented, which can be reduced to the nonlocal GP equation with self-induced PT-symmetric potential. We derive some novel non-autonomous soliton solutions of nonlocal GP equation via inverse scattering and similarity transformations. These obtained solutions have singularities and are not the standard soliton solutions, here they can be called soliton-like(SL) solutions. Then we consider some controllable behaviors of these non-autonomous wave solutions. The obtained results are different from the solutions of the local nonlinear GP equation. Some propagation phenomena are produced through manipulating non-autonomous SL waves, which can present the potential applications to the soliton wave phenomena in nonlocal wave models.

Optical solitons of the (1 + 1)-dimensional higher-order nonlinear Schrödinger equations with [formula omitted]-symmetric potentials

The (1+1)-dimensional higher-order nonlinear Schrödinger equations with PT-symmetric potentials are presented. Soliton solutions form in different nonlinear media under the balance between the second-order diffraction and the fourth-order diffraction, PT-symmetric potentials with higher-order nonlinearities, such as the cubic-quintic nonlinearities, quintic-septimal nonlinearities and cubic-quintic-septimal nonlinearities. The coefficients of the fourth-order diffraction and second-order diffraction determine the propagation constant of soliton, and the coefficients of the gain or loss distribution, the second-order diffraction and the fourth-order diffraction decide the phase of soliton.

Dynamical evolution of optical solitons in the (1+1)-dimensional quintic-septimal media with [formula omitted]-symmetric potentials

The (1+1)-dimensional constant-coefficient and variable-coefficient quintic-septimal nonlinear Schrödinger equations with PT-symmetric potentials are studied. Two kinds of explicit soliton solutions for constant-coefficient equation are derived. Based on the one-to-one relation between variable-coefficient and constant-coefficient equations with these solutions of constant-coefficient equation, two kinds of explicit soliton solutions for variable-coefficient equation are obtained. In the homogeneous case, the formation of solitons is decided by the diffraction and septimal nonlinear coefficients. The phase of soliton exists a jump from the negative to positive values at the position of center.

Dynamics of some novel breather solutions and rogue waves for the PT-symmetric nonlocal soliton equations

A generalized nonlocal nonlinear Hirota (GNNH) equation with variable coefficients is presented, which can be reduced into the nonlocal Hirota equation with the self-induced PT-symmetric potential. Especially, the nonlocal Gross–Pitaevskii (NGP) equation with the self-induced PT-symmetric potential is derived from the GNNH equation. Then, we obtain some novel non-autonomous breather solutions and rogue waves of GNNH equation via similarity and Hirota methods, and consider some controllable behaviors of these non-autonomous wave solutions. Furthermore, some properties of the non-autonomous rational (NR) waves are investigated analytically for the NGP equation. The trajectories of peaks and depressions of the non-autonomous rogue waves are produced by means of analytical method, and the dynamical stabilities of the NR solution are derived through the numerical method. The obtained results are different from the solutions of the local nonlinear equations. Some different propagation phenomena can also be produced through manipulating non-autonomous rogue waves, which can present the potential applications for the rogue wave phenomena in nonlocal wave models.

Evolution of vortex and quadrupole solitons in the complex potentials with saturable nonlinearity

We investigate the evolution of vortex and quadrupole solitons in the parity-time (PT)-symmetric optical lattices with self-focusing saturable nonlinearity. It is found that the saturable parameter plays a significant role on the existence and stability of vortex solitons. There exists a threshold for saturable parameter, above which the single-charged vortex solitons can be stable. The strength of gain–loss component of the PT-symmetric potential can affect the existence and stability of vortex solitons dramatically. We find that the single-charged vortex solitons can exist even at the phase transition point of the PT-symmetric potential, but diffuse fast during propagation. In addition, it is also found that the direction of the angular momentum has little effect on the properties of vortex solitons. Dramatically, the stable regions of out-of-phase quadrupole solitons are broader than that of the single-charged vortex solitons, while the stable regions of in-phase quadrupole solitons are narrower than that of the single-charged vortex solitons.

Integrability conditions and solitonic solutions of the nonlinear Schrödinger equation with generalized dual-power nonlinearities, PT-symmetric potentials, and space- and time-dependent coefficients

We consider a generalized nonlinear Schrödinger equation with dual power law nonlinearities, complex potential, and position- and time-dependent strengths of dispersion and nonlinearities. Using a standard similarity transformation, we obtain the integrability conditions and solitonic solutions of this equation by mapping it to its homogeneous version. Using a modified similarity transformation, where a solution of the homogeneous equation, which we denote as a seed, enters also in the transformation operator, a wider range of exact solutions is obtained including cases with complex potentials. We apply these two transformations to obtain two exact solitonic solutions of the homogeneous nonlinear Schrödinger equation, which are derived here for the first time for a general power of the nonlinearities, namely the flat-top soliton and tanh solution. We discuss and derive explicit solutions to the experimentally relevant cases associated with parabolic and PT-symmetric potentials.

Gauss-type solitons of (1+1)-dimensional cubic-quintic-septimal nonlinear Schrödinger equation with Gauss-type [formula omitted]-symmetric potentials

The Gauss-type PT-symmetric potentials acting on cubic-quintic-septimal nonlinear media are studied, and two kinds of Gauss-type solitons in these media are analytically derived. The comparison of form characteristics between two kinds of Gauss-type solitons is discussed, including the real part, imaginary part and intensity of Gauss-type solitons. Moreover, based on the eigenvalue method, the linear stability analysis for solutions is given, and then utilizing numerical simulation with the split-step Fourier method, the stability of Gauss-type solitons in different nonlinear media is further checked.

A class of exactly solvable rationally extended non-central potentials in two and three dimensions

We start from a seven parameter (six continuous and one discrete) family of non-central exactly solvable potentials in three dimensions and construct a wide class of ten parameters (six continuous and four discrete) family of rationally extended exactly solvable non-central real as well as PT symmetric complex potentials. The energy eigenvalues and the eigenfunctions of these extended non-central potentials are obtained explicitly and it is shown that the wave eigenfunctions of these potentials are either associated with the exceptional orthogonal polynomials or some type of new polynomials which can be further re-expressed in terms of the corresponding classical orthogonal polynomials. Similarly, we also construct a wide class of rationally extended exactly solvable non-central real as well as complex PT-invariant potentials in two dimensions.

Stability and soliton solutions for a parity-time-symmetric vector nonlinear Schrödinger system

In this paper, we consider the two-coupled nonlinear Schrodinger equations with parity-time-symmetric potential in the presence of four-wave mixing. We construct the soliton solutions for the vector nonlinear Schrodinger equations with some PT-symmetric potentials. Then the linear-stability spectrum for solitary waves is studied. Moreover, soliton solutions in high dimensional case are also considered.

Double-hump solitons in fractional dimensions with a 𝒫𝒯-symmetric potential.

We investigate the properties of double-hump solitons supported by the nonlinear Schrödinger equation featuring a combination of parity-time symmetry and fractional-order diffraction effect. Two classes of nonlinear states, i.e., out-of-phase and in-phase solitons are found. Each class contains two families of solitons originating from the same linear mode in both focusing and defocusing nonlinear Kerr media. The critical phase-transition point increases monotonously with increasing Lévy index. For strong gain and loss, out-of-phase solitons in focusing media are stable in a wide parameter window and are almost completely unstable in media with a defocusing nonlinearity. The stability of in-phase solitons is opposite to that of out-of-phase solitons. In-phase solitons in defocusing media are stable in their entire existence domains provided that the gain-loss strength is below a critical value. Meanwhile, the stability region shrinks with the decrease of Lévy index. We, thus, put forward the first example of spatial solitons in fractional dimensions with a parity-time symmetry.

Effect of PT symmetry on nonlinear waves for three-wave interaction models in the quadratic nonlinear media.

We study the three-wave interaction that couples an electromagnetic pump wave to two frequency down-converted daughter waves in a quadratic optical crystal and PT-symmetric potentials. PT symmetric potentials are shown to modulate stably nonlinear modes in two kinds of three-wave interaction models. The first one is a spatially extended three-wave interaction system with odd gain-and-loss distribution in the channel. Modulated by the PT-symmetric single-well or multi-well Scarf-II potentials, the system is numerically shown to possess stable soliton solutions. Via adiabatical change of system parameters, numerical simulations for the excitation and evolution of nonlinear modes are also performed. The second one is a combination of PT-symmetric models which are coupled via three-wave interactions. Families of nonlinear modes are found with some particular choices of parameters. Stable and unstable nonlinear modes are shown in distinct families by means of numerical simulations. These results will be useful to further investigate nonlinear modes in three-wave interaction models.

Symmetric and asymmetric solitons supported by a 𝒫𝒯-symmetric potential with saturable nonlinearity: bifurcation, stability and dynamics.

The symmetry breaking bifurcation of solitons in an optical waveguide with focusing saturable nonlinearity and parity-time (𝒫𝒯)-symmetric complex-valued external potentials is investigated. As the soliton power increases, it is found that the branches of asymmetric solitons split off from the base branches of 𝒫𝒯-symmetric fundamental soliton. The bifurcation diagrams, consisting essentially of the propagation constants of optical solitons, indicate that symmetric fundamental and multipole solitons, as well as asymmetric solitons can exist. The stabilities and the dynamics characteristics of solitons are comprehensively investigated. We find the different instability scenarios of the symmetric solitons, but the symmetry breaking bifurcation is caused only by the onset of instability of the symmetric fundamental solitons. This result is further confirmed by the numerical examples with the different saturable nonlinearity parameters. In particular, we find that the soliton power and the stability of soliton at the bifurcation points are significantly changed by varying the strength of the saturable nonlinearities. These results provide additional way to control symmetry breaking bifurcations in 𝒫𝒯-symmetric optical waveguide.

Two-dimensional solitons in cubic-saturable media with [formula omitted]-symmetric lattices

In this paper, we discuss the existence of lattice solitons supported by cubic-saturable nonlinearity in the framework of nonlinear Schrödinger (NLS) equation with external potentials such as parity-time-symmetric lattices with and without defects by using the pseudo-spectral renormalization method. Linear and nonlinear stability properties of the lattice solitons centered on the maximum of the PT-symmetric potential are investigated in detail.

Soliton structures in the (1+1)-dimensional Ginzburg–Landau equation with a parity-time-symmetric potential in ultrafast optics**Project supported by the National Natural Science Foundation of China (Grant No. 11674036), the Beijing Youth Top-notch Talent Support Program, China (Grant No. 2017000026833ZK08), and the Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications) (Grant Nos. IPOC2016ZT04 and

In this paper, the (1+1)-dimensional variable-coefficient complex Ginzburg–Landau (CGL) equation with a parity-time (PT) symmetric potential U(x) is investigated. Although the CGL equations with a PT-symmetric potential are less reported analytically, the analytic solutions for the CGL equation are obtained with the bilinear method in this paper. Via the derived solutions, some soliton structures are presented with corresponding parameters, and the influences of them are analyzed and studied. The single-soliton structure is numerically verified, and its stability is analyzed against additive and multiplicative noises. In particular, we study the soliton dynamics under the impact of the PT-symmetric potential. Results show that the PT-symmetric potential plays an important role for obtaining soliton structures in ultrafast optics, and we can design fiber lasers and all-optical switches depending on the different amplitudes of soliton-like structures.

N-Fold Darboux Transformation for the Nonlocal Nonlinear Schrödinger (NNLS) Equation with the Self-Induced PT-Symmetric Potential

The Darboux transformation (DT) method is studied in a lot of local equations, but there are few of work to solve nonlocal equations by DT. In this letter, we solve the nonlocal nonlinear Schrödinger equation (NNLSE) with the self-induced PT-symmetric potential by DT. Then the N-fold DT of NNLSE is derived with the help of the gauge transformation between the Lax pairs. Then we derive some novel exact solutions including the bright soliton, breather wave soliton. In particularly, the dynamic features of one-soliton, two-soliton, three-soliton solutions and the elastic interactions between the two solitons are displayed.

The spectrum of the Hamiltonian with a PT-symmetric periodic optical potential

We give a complete description, provided with a mathematical proof, of the shape of the spectrum of the Hill operator with potential [Formula: see text], where [Formula: see text] We prove that the second critical point [Formula: see text], after which the real parts of the first and second bands disappear, is a number between [Formula: see text] and [Formula: see text]. Moreover, we prove that [Formula: see text] is the degeneration point for the first periodic eigenvalue. Besides, we give a scheme by which one can find arbitrary precise value of the second critical point as well as the [Formula: see text]th critical points after which the real parts of the [Formula: see text]th and [Formula: see text]th bands disappear, where [Formula: see text]

Bright-dark and dark-dark solitons in coupled nonlinear Schrödinger equation with PT-symmetric potentials.

We explore different nonlinear coherent structures, namely, bright-dark (BD) and dark-dark (DD) solitons in a coupled nonlinear Schrödinger/Gross-Pitaevskii equation with defocusing/repulsive nonlinearity coefficients featuring parity-time ( PT)-symmetric potentials. Especially, for two choices of PT-symmetric potentials, we obtain the exact solutions for BD and DD solitons. We perform the linear stability analysis of the obtained coherent structures. The results of this linear stability analysis are well corroborated by direct numerical simulation incorporating small random noise. It has been found that there exists a parameter regime which can support stable BD and DD solitons.