Symmetric Complex Potentials Research Articles

Stability of bright solitons in optical system supported by cubic and quintic nonlinearities with PT -symmetric quartic harmonic complex potential

This work analyzed the existence and stability of optical waves in linear and nonlinear media with self-focusing cubic-quintic nonlinearity by way of interactions of nonlinearity, spatial diffraction, and parity-time ( PT ) symmetric complex potential defined by quartic harmonic real potential and Gaussian imaginary distribution. We demonstrated that the order of nonlinear function, the width and depth of the real potential, and the coefficient of diffraction modify the span of the complex potential parameter onto which PT -symmetry breakdown occurs. The region of stable PT -symmetry in the self-focusing medium improves as the order of the nonlinear function increases, as demonstrated by comparing the soliton solutions with cubic and quintic nonlinearities. We addressed the nonlinear evolution of the stable and unstable PT solitons under propagation. The linear stability of bright solitons is extensively studied and established the conditions for stable soliton for both PT− symmetric and broken PT− symmetric regions.

Beam propagation in cubic-quintic nonlinear optical system supported by modified parity-time symmetric Rosen-Morse complex potential.

The present study explores the stability and persistence of nonlinear waves in self-focusing cubic-quintic media employing couplings of nonlinearity, spatial diffraction, and the parity-time symmetric Rosen-Morse complex field. Here, we discover that a system supported by Rosen-Morse potential can develop eigenmodes but does not accommodate stable soliton solutions for any potential parameters due to the non-vanishing terms in the imaginary component of potential. The study expands by modifying Rosen-Morse potential and discovers that the region of sustained PT-symmetry in the self-focusing material is enhanced and influenced by the order of the nonlinear function, spectral filtering, and gain/loss in the system. Stable soliton conditions for both broken PT-symmetric and PT-symmetric regions are established by the linear stability analysis using numerical simulations. Nonlinear propagation of the beam in the modified PT system is explored and identifies that stable beam propagation is possible only if the system is supported by the field, which is below the threshold imaginary potential.

Nonlinear eigen modes in optical media supported by cubic and quintic nonlinearities with Parity-Time symmetric hyperbolic complex potential

This work explored optical waves in different nonlinear media, specifically in self-focusing cubic–quintic media, and investigated how these waves evolve and maintain their characteristics under the influence of various factors such as nonlinear interactions, diffraction, and Parity-Time (PT) symmetric complex field with hyperbolic real and imaginary distributions. We explored how the range of the complex field parameter over which PT-symmetry collapse occurs is modified by the order of the nonlinear function, modulation strength that makes up the potential, the separation between peaks, and the width of the PT-symmetric field. A comparison of the soliton solutions with Kerr and quintic nonlinearities clarifies that the span of sustainable PT-symmetry in the self-focusing medium expands with the higher order nonlinear function. We discussed the propagation-related nonlinear evolutions of unstable and robust PT solitons. The conditions necessary for sustained soliton in PT-symmetric and broken PT-symmetric domains were identified by the study of the linear stability of solitons.

Soliton dynamics and stability in the ABS spinor model with a PT -symmetric periodic complex potential

We investigate the effects on solitons dynamics of introducing a PT -symmetric complex potential in a specific family of the cubic Dirac equation in (1+1)-dimensions, called the Alexeeva–Barashenkov–Saxena model. The potential is introduced taking advantage of the fact that the nonlinear Dirac equation admits a Lagrangian formalism. As a consequence, the imaginary part of the potential, associated with gains and losses, behaves as a spatially periodic damping (changing from positive to negative, and back) that acts at the same time on the two spinor components. A collective coordinates (CCs) theory is developed by making an ansatz for a moving soliton where the position, rapidity, momentum, frequency, and phase are all functions of time. We consider the complex potential as a perturbation and verify that numerical solutions of the equation of motions for the CCs are in agreement with simulations of the nonlinear Dirac equation. The main effect of the imaginary part of the potential is to induce oscillations in the charge and energy (they are conserved for real potentials) with the same frequency and phase as the momentum. We find long-lived solitons even with very large charge and energy oscillations. Additionally, we extend to the nonlinear Dirac equation an empirical stability criterion, previously employed successfully in the nonlinear Schrödinger equation.

PT-symmetry rules applied to a class of real potentials

Extending the functional space to complex eigenfunctions R J Lombard et al (2022, Rom. J. Phys. 67, 104), we have shown that infinitely negative potentials at large distances admit finite energy states. The used techniques are similar to the ones applied in the case of PT symmetric complex potentials with real eigenvalues. We present the lowest part of the spectra for −∣x∣ n potentials with 4 ≤ n ≤ 8. We also discuss the norm and orthogonality of the wave functions.

Two-dimensional fractional [formula omitted]-symmetric cubic–quintic NLS equation: Double-loop symmetry breaking bifurcations, ghost states and dynamics

In this paper, we address the double-loop symmetry breaking bifurcations of optical vortex solitons in the two-dimensional (2D) fractional cubic–quintic nonlinear Schrödinger (FCQNLS) equation with an imprinted partially parity-time (PPT) symmetric complex potential. The continuous branches of asymmetric solitons bifurcate from the fundamental one as the power exceeds some critical value. It is found that the symmetry breaking point is exactly the point at which the fundamental symmetric soliton is destabilized. Intriguingly, the branches of asymmetry solitons (alias ghost states) are existing with complex conjugate propagation constants, which is solely in fractional media. Moreover, the branch of asymmetric solitons crosses the branch of fundamental symmetric ones when it reaches the second critical value, and eventually merges back into it at the third critical power point. The imaginary part of the propagation constant also exhibits a circular shape. And the stability of the fundamental symmetric soliton is restored. In addition, the stability of two branches of asymmetric solitons differs from each other. The reason generating this phenomenon is analyzed in detail. Besides, the branch of dipole solitons for the first excited state is also discussed numerically, which is always stable in the process of symmetry breaking. The stability and dynamics of symmetric (fundamental and the first excited ones) and asymmetric solitons are explored via the linear stability analysis, direct simulation, modulation as well as adiabatic excitation. These results will provide some theoretical basis for the study of spontaneous symmetry breaking phenomena and related experiments in fractional media with PPT-symmetric potentials.

Self-bound states induced by the Lee–Huang–Yang effect in non-$$\mathcal{PT}\mathcal{}$$-symmetric complex potentials

Self-bound states induced by the Lee–Huang–Yang effect in non-$$\mathcal{PT}\mathcal{}$$-symmetric complex potentials

Self-Bound States Induced by the Lee-Huang-Yang Effect in Non-Pt -Symmetric Complex Potentials

Self-Bound States Induced by the Lee-Huang-Yang Effect in Non-Pt -Symmetric Complex Potentials

Supersymmetry of 𝒫𝒯-symmetric tridiagonal Hamiltonians

We extend the study of supersymmetric tridiagonal Hamiltonians to the case of non-Hermitian Hamiltonians with real or complex conjugate eigenvalues. We find the relation between matrix elements of the non-Hermitian Hamiltonian [Formula: see text] and its supersymmetric partner [Formula: see text] in a given basis. Moreover, the orthogonal polynomials in the eigenstate expansion problem attached to [Formula: see text] can be recovered from those polynomials arising from the same problem for [Formula: see text] with the help of kernel polynomials. Besides its generality, the developed formalism in this work is a natural home for using the numerically powerful Gauss quadrature techniques in probing the nature of some physical quantities such as the energy spectrum of [Formula: see text]-symmetric complex potentials. Finally, we solve the shifted [Formula: see text]-symmetric Morse oscillator exactly in the tridiagonal representation.

Solitons in Kerr media with two-dimensional non-parity-time-symmetric complex potentials

Our results indicate that continuous soliton families can exist and be stable in Kerr media with two-dimensional (2D) non-parity-time (PT)-symmetric complex potentials. There are several discrete eigenvalues in the linear spectra of these complex potentials. Fundamental solitons bifurcate from the largest discrete eigenvalue while the dipole solitons bifurcate from the second- or third- largest discrete eigenvalue. We further find that eigenvalues of the soliton linear-stability spectra emerge as complex conjugate pairs. The effect of different parameters of the complex potentials on soliton stability is discussed in detail. Moreover, we study the transverse energy flow vector of the solitons in these complex potentials.

Influence of [formula omitted]-symmetric complex potentials on the decoupling mechanism in quantum transport processes

We consider one system in which the terminal dots of a one-dimensional quantum-dot chain couple equally to the left and right leads and study the influence of PT-symmetric complex potentials on the quantum transport process. It is found that in the absence of the complex potentials, remarkable decoupling and antiresonance phenomena are able to co-occur in the transport process. For the chains with odd(even) dots, their even(odd)-numbered molecular states decouple from the leads. Meanwhile, antiresonance occurs at the positions of the even(odd)-numbered eigenenergies of the sub-chains without terminal dots. When the PT-symmetric complex potentials are applied on the terminal dots, the decoupling phenomenon is found to transform into the Fano antiresonance. These results can helpful in understanding the quantum transport modified by the PT symmetry in non-Hermitian discrete systems.

Nonlocal solitons supported by non-parity-time-symmetric complex potentials

We report on the existence and stability of fundamental and out-of-phase dipole solitons in nonlocal focusing Kerr media supported by one-dimensional non-parity-time (PT)-symmetric complex potentials. These fundamental and dipole solitons bifurcate from different discrete eigenvalues in the linear spectra. Below the phase transition of the non-PT-symmetric complex potentials, these solitons are stable in the low power region. While above the phase transition, they are stable in the moderate power region. The eigenvalues in linear-stability spectra of solitons appear as conjugation pairs (δ, δ*). The transverse power flow and the nonlinear contribution to refractive index are asymmetric functions. Moreover, the degree of nonlocality can also influence the stability of these solitons.

Transmission in a dimerized chain influenced by -symmetric potentials

We investigate the effect of parity-time ()-symmetric complex potentials on the transmission in a non-Hermitian system, which consists of an infinite dimerized chain and two side-coupled defects with -symmetric complex on-site potentials. It is found that the -symmetric imaginary potentials have pronounced effects on the transmission behavior of such a system, including the shift of the normal antiresonance and the occurrence or disappearance of the Fano resonance. These phenomena depend significantly on the coupling manners and strengths between the defects and the dimerized chain. Our study can be helpful in understanding the influences of the symmetry of the dimerized systems on transmission properties.

Effects of a non-Hermitian potential on the Landau quantization

In this work, we solve the time-independent Schrödinger equation for a Landau system modulated by a non-Hermitian Hamiltonian. The system consists of a spinless particle in a uniform magnetic field submitted to action of a non-[Formula: see text] symmetric complex potential. Although the Hamiltonian is neither Hermitian nor [Formula: see text]-symmetric, we find that the Landau problem under study exhibits an entirely real energy spectrum.

Eigenenergies and quantum transport properties in a non-Hermitian quantum-dot chain with side-coupled dots

We study the eigenenergies of a one-dimensional quantum-dot chain with its each dot side coupling to two additional dots. It is found that when the $\mathcal{P}T$-symmetric complex potentials are introduced to the side-coupled dots, the eigenlevel degeneracy is broken. However, further increasing the complex potentials induces a kind of two-degree degeneracy of the eigenlevels. This is accompanied by the $\mathcal{P}T$-symmetry breaking, with the appearance of the complex part of the eigenlevels. These changes exactly affect the quantum transport properties of the chain. First, in the case of weak $\mathcal{P}T$-symmetric complex potentials, a group of transmission function peaks arise at the center of the transmission function spectrum. When the eigenlevel degeneracy takes place, the degenerated eigenlevels decouple from the leads and the corresponding peaks disappear in the transmission function spectra. We believe that this work provides helpful information for a better understanding of the eigenenergies and quantum transport properties in non-Hermitian systems.

Global search for localised modes in scalar and vector nonlinear Schrödinger-type equations

We present a new approach for search of coexisting classes of localised modes admitted by the repulsive (defocusing) scalar or vector nonlinear Schr\"odinger-type equations. The approach is based on the observation that generic solutions of the corresponding stationary system have singularities at finite points on the real axis. We start with establishing conditions on the initial data of the associated Cauchy problem that guarantee the formation of a singularity. Making use of these sufficient conditions, we identify the bounded, nonsingular, solutions --- and then classify them according to their asymptotic behaviour. To determine the bounded solutions, a properly chosen space of initial data is scanned numerically. Due to asymptotic or symmetry considerations, we can limit ourselves to a one- or two-dimensional space. For each set of initial conditions we compute the distances $X^{\pm}$ to the nearest forward and backward singularities; large $X^+$ or $X^-$ indicate the proximity to a bounded solution. We illustrate our method with the Gross-Pitaevskii equation with a $\PT$-symmetric complex potential, a system of coupled Gross-Pitaevskii equations with real potentials, and the Lugiato-Lefever equation with normal dispersion.

Continuous families of solitary waves in non-symmetric complex potentials: A Melnikov theory approach

The existence of stationary solitary waves in symmetric and non-symmetric complex potentials is studied by means of Melnikov’s perturbation method. The latter provides analytical conditions for the existence of such waves that bifurcate from the homogeneous nonlinear modes of the system and are located at specific positions with respect to the underlying potential. It is shown that the necessary conditions for the existence of continuous families of stationary solitary waves, as they arise from Melnikov theory, provide general constraints for the real and imaginary part of the potential, that are not restricted to symmetry conditions or specific types of potentials. Direct simulations are used to compare numerical results with the analytical predictions, as well as to investigate the propagation dynamics of the solitary waves.

Vector solitons in nonparity-time-symmetric complex potentials.

The existence and stability of vector solitons in non-parity-time (PT)-symmetric complex potentials are investigated. We study the vector soliton family, in which the propagation constants of the two components are different. It is found that vector solitons can be stable below and above the phase transition of the non-PT-symmetric complex potentials. Below the phase transition, vector solitons are stable in the low power region. Above the phase transition, there are two continuous stable intervals in the existence region. The profiles of two components of these vector solitons show the asymmetry and we also study the transverse power flow in the two components of these vector solitons in the non-PT-symmetric complex potentials.

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.

Effect of PT symmetry in a parallel double-quantum-dot structure

We present a comprehensive analysis about the effect of $\mathcal{PT}$-symmetric complex potentials on quantum transport through one parallel coupled double-quantum-dot structure. It is found that such potentials are able to eliminate the decoupling phenomenon while inducing the Fano effect, even under the situation of uniform dot-lead coupling. Moreover, the Fano line shape in the transmission function spectrum is opposite to that that arises from the detuning of the dot levels. We ascertain that this work can be helpful in understanding the quantum transport properties in the quantum systems with $\mathcal{PT}$ symmetry.