Complex PT-symmetric Potential Research Articles

Physics-informed Neural Network method for predicting soliton dynamics supported by complex PT-symmetric potentials

Abstract We examine the deep learning technique referred to as the physics-informed neural network method for approximating non-linear Schrödinger equation under considered parity time symmetric potentials and obtaining multifarious soliton solutions. For the first time, neural networks founded principally physical information are adopted to figure out the solution the examined non-linear partial differential equation and generate six different types of soliton solutions, which are basic, dipole, tripole, quadruple, pentapole and sextupole solitons we consider. We make comparisons between the predicted and actual soliton solutions to see whether deep learning is capable of seeking the solution the partial differential equation described before. We may assess whether physics-informed neural network is capable of effectively providing approximate soliton solutions through the evaluation of squared error between the predicted and numerical results. Besides, we also scrutinize how different activation mechanisms and network architectures impact the capability of selected deep learning technique works. Through the findings we can prove that the neural networks model we established can be utilized to accurately and effectively approximate non-linear Schrödinger equation under consideration and predict the dynamics of soliton solution.

An efficient method to solve the Schrödinger equation with complex PT-symmetric potentials

In this work, we present a simple and efficient method to compute numerically the eigenvalues of complex PT-symmetric Hamiltonian. Numerous works have been devoted to such Hamiltonians since the discovery that they admit totally or partially real spectra. To our knowledge, the method we are advocating has not been used in this context. Besides the determination of real eigenvalues, it allows us to observe the symmetry breaking and to calculate the imaginary parts of the energy.

Dynamics of stable solitons in Complex Ginzburg–Landau equation with [formula omitted]-symmetric Gaussian potential

In this paper, we investigate the stable propagation of soliton in Complex Ginzburg–Landau (CGL) equation with self focusing nonlinear mode in the presence of PT-symmetric Gaussian potential. In our model, we find the required condition to obtain the stable solution is that the value of spectral filtering is negative and the positive values of diffraction, Kerr nonlinearity and nonlinear gain/loss. By satisfying this condition, one can manipulate the stable propagation, intensity and power conservation of the soliton by simply varying the strength of the imaginary part of the complex PT-symmetric potential. In other words, the dynamics of the soliton depend on the strength of the imaginary part of potential irrespective of the value of propagation constant.

Data-driven rogue waves and parameters discovery in nearly integrable [formula omitted]-symmetric Gross–Pitaevskii equations via PINNs deep learning

In this paper, we explore the forward and inverse problems for the generalized Gross–Pitaevskii (GP) equation with complex PT-symmetric potentials via the deep physics-informed neural networks (PINNs). The data-driven rogue waves (RWs) are mainly studied in the forward problem, where the comparisons between the data-driven RWs and numerical ones via the spectral method are used to present the PINNs solution accuracies. Besides, we also focus on the influences of several critical factors (e.g., the depths of neural networks, numbers of training points) on the performance of the PINNs algorithm. Finally, the inverse problem is also investigated such that the system parameters can be identified from the training data. The results obtained in this paper can be useful to further understand the neural networks on rogue wave structures in the nearly integrable PT-symmetric nonlinear wave systems.

Rogue wave formation and interactions in the defocusing nonlinear Schrödinger equation with external potentials

The defocusing nonlinear Schrödinger (NLS) equation has no the modulational instability, and was not found to possess the rogue wave (RW) phenomenon up to now. In this paper, we firstly investigate some novel nonlinear wave structures in the defocusing NLS equation with real-valued time-dependent and time-independent potentials such that the stable new RWs and W-shaped solitons are found, respectively. Moreover, the interactions of two or three RWs are explored such that the RWs with higher amplitudes are generated in the defocusing NLS equation with real-valued time-dependent potentials. Finally, we study the defocusing NLS equation with complex PT-symmetric potentials such that some RWs and W-shaped solitons are also found. These novel results will be useful to design the related physical experiments to generate the RW phenomena and W-shaped solitons in the case of defocusing nonlinear interactions, and to apply them in the related fields of nonlinear or even linear sciences.

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.

The analytical transfer matrix method for PT-symmetric complex potential

We have extended the analytical transfer matrix (ATM) method to solve quantum mechanical bound state problems with complex PT-symmetric potentials. Our work focuses on a class of models studied by Bender and Jones, we calculate the energy eigenvalues, discuss the critical values of g and compare the results with those obtained from other methods such as exact numerical computation and WKB approximation method.

Exact localized solutions of (1 + 1)-dimensional nonlinear Schrödinger equation with complex 𝒫𝒯 symmetric potentials and power-law nonlinearity

Using the concept of deformed supersymmetry, we obtain exact localized solutions of nonlinear Schrödinger equation (NLSE) with a number of [Formula: see text] symmetric potentials and [Formula: see text]th power-law nonlinearity. Linear stability of these solutions for focusing as well as de-focusing nonlinearity has been examined for various values of the nonlinearity parameter [Formula: see text].

Parametric symmetries in exactly solvable real and PT symmetric complex potentials

In this paper, we discuss the parametric symmetries in different exactly solvable systems characterized by real or complex PT symmetric potentials. We focus our attention on the conventional potentials such as the generalized Pöschl Teller (GPT), Scarf-I, and PT symmetric Scarf-II which are invariant under certain parametric transformations. The resulting set of potentials is shown to yield a completely different behavior of the bound state solutions. Further, the supersymmetric partner potentials acquire different forms under such parametric transformations leading to new sets of exactly solvable real and PT symmetric complex potentials. These potentials are also observed to be shape invariant (SI) in nature. We subsequently take up a study of the newly discovered rationally extended SI potentials, corresponding to the above mentioned conventional potentials, whose bound state solutions are associated with the exceptional orthogonal polynomials (EOPs). We discuss the transformations of the corresponding Casimir operator employing the properties of the so(2, 1) algebra.

Transparency of the complex PT-symmetric potentials for coherent injection

Two port s-matrix for a complex PT-symmetric potential may have uni-modular eigenvalues. If this happens for all energies, there occurs a perfect emission of waves at both ends. We call this phenomenon transparency which is distinctly different from coherent perfect absorption with or without lasing. Using the versatile PT-symmetric complex Scarf II (scattering) potential, we demonstrate analytically that the transparency can occur regardless of whether PT-symmetry is unbroken or broken or if there are only scattering states. In these three cases, for a given value of the strength of the real part, the strength of the imaginary part |V2| of the potential lies in (0,Vα),(Vα,Vβ) and (0,Vβ) respectively. Several other numerically solved potentials also support our findings.

Accidental crossings of eigenvalues in the one-dimensional complex PT-symmetric Scarf-II potential

So far, the well known two branches of real discrete spectrum of complex PT-symmetric Scarf-II potential are kept isolated. Here, we suggest that these two need to be brought together as doublets: E±n(λ) with n=0,1,2… . Then if strength (λ) of the imaginary part of the potential is varied smoothly some pairs of real eigenvalue curves can intersect and cross each other at λ=λ⁎; this is unlike one-dimensional Hermitian potentials. However, we show that the corresponding eigenstates at λ=λ⁎ are identical or linearly dependent denying degeneracy in one dimension, once again. Other pairs of eigenvalue curves coalesce to complex-conjugate pairs completing the scenario of spontaneous breaking of PT-symmetry at λ=λc. To re-emphasize, sharply at λ=λ⁎ and λc, two real eigenvalues coincide, nevertheless their corresponding eigenfunctions become identical or linearly dependent and the Hamiltonian looses diagonalizability.

A new solvable complex PT-symmetric potential

We propose a new solvable one-dimensional complex PT-symmetric potential as V(x)=igsgn(x)|1−exp⁡(2|x|/a)| and study the spectrum of H=−d2/dx2+V(x). For smaller values of a,g<1, there is a finite number of real discrete eigenvalues. As a and g increase, there exist exceptional points (EPs), gn (for fixed values of a), causing a scarcity of real discrete eigenvalues, but there exists at least one. We also show these real discrete eigenvalues as poles of reflection coefficient. We find that the energy-eigenstates ψn(x) satisfy (1): PTψn(x)=1ψn(x) and (2): PTψEn(x)=ψEn⁎(x), for real and complex energy eigenvalues, respectively.

Reciprocity and unitarity in scattering from a non-Hermitian complex PT-symmetric potential

In quantum scattering, Hermiticity is necessary for both reciprocity and unitarity. Reciprocity means that both reflectivity (R) and transmitivity (T) are insensitive to the direction of incidence of a wave (particle) at a scatterer from left/right. Unitarity means that R+T=1. In scattering from non-Hermitian PT-symmetric structures the (left/right) handedness (non-reciprocity) of reflectivity is known to be essential and unitarity remains elusive so far. Here we present a surprising occurrence of both reciprocity and unitarity in scattering from a complex PT-symmetric potential. In special cases, we show that this potential can even become invisible (R=0, T=1) from both left and right sides. We also find that this potential in a parametric regime enjoys a pseudo-unitarity of the type $T+\sqrt{R_{left}R_{right}}=1$.

Eigenvalue problems for the complex PT-symmetric potential

The spectrum of complex PT-symmetric potential, $V(x)=igx$, is known to be null. We enclose this potential in a hard-box: $V(|x| \ge 1) =\infty $ and in a soft-box: $V(|x|\ge 1)=0$. In the former case, we find real discrete spectrum and the exceptional points of the potential. The asymptotic eigenvalues behave as $E_n \sim n^2.$ The solvable purely imaginary PT-symmetric potentials vanishing asymptotically known so far do not have real discrete spectrum. Our solvable soft-box potential possesses two real negative discrete eigenvalues if $|g|<(1.22330447)$. The soft-box potential turns out to be a scattering potential not possessing reflectionless states.

A new class of PT-symmetric Hamiltonians with real spectra

We investigate complex PT-symmetric potentials, associated with quasi-exactly solvable non-Hermitian models involving polynomials and a class of rational functions. We also look for special solutions of intertwining relations of SUSY quantum mechanics providing a partnership between a real and a complex PT-symmetric potential of the kind mentioned above. We investigate conditions sufficient to ensure the reality of the full spectrum or, for the quasi-exactly solvable systems, the reality of the energy of the finite number of levels.

PT-symmetric sextic potentials

The family of complex PT-symmetric sextic potentials is studied to show that for various cases the system is essentially quasi-solvable and possesses real, discrete energy eigenvalues. For a particular choice of parameters, we find that under supersymmetric transformations the underlying potential picks up a reflectionless part.