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Recent progress on nonlinear properties of parity-time ($\cal PT$-) symmetric systems is comprehensively reviewed in this article. $\cal PT$ symmetry started out in non-Hermitian quantum mechanics, where complex potentials obeying $\cal PT$…
We present a systematic analysis of the stationary regimes of nonlinear parity-time(PT) symmetric laser composed of two coupled fiber cavities. We find that power-dependent nonlinear phase shifters broaden regions of existence of both…
A fundamental problem in the theory of PT-invariant quantum systems is to determine whether a given system `respects' this symmetry or not. If not, the system usually develops non-real eigenvalues. It is shown in this contribution how to…
The Hamiltonian for a PT-symmetric chain of coupled oscillators is constructed. It is shown that if the loss-gain parameter $\gamma$ is uniform for all oscillators, then as the number of oscillators increases, the region of unbroken…
The eigenoscillations in a two-dimensional dimer made of gain and lossy nanoparticles were investigated within exact analytical approach. It was shown that there are eigenoscillations for which all Joule losses are exactly compensated by…
In the present work, we consider a prototypical example of a PT-symmetric Dirac model. We discuss the underlying linear limit of the model and identify the threshold of the PT-phase transition in an analytical form. We then focus on the…
We introduce a dual-core system with double symmetry, one between the cores, and one along each core, imposed by the spatial modulation of local nonlinearity in the form of two tightly localized spots, which may be approximated by a pair of…
We investigate the dynamics of parity- and time-reversal (PT ) symmetric two-energy-level atoms in the presence of two optical and a radio-frequency (rf) fields. The strength and relative phase of fields can drive the system from unbroken…
Parity-time (PT) symmetry and broken in micro/nano photonic structures have been investigated extensively as they bring new opportunities to control the flow of light based on non-Hermitian optics. Previous studies have focused on the…
Dynamical symmetries of Hamiltonians quantized models of discrete non-linear Schroedinger chain (DNLS) and of Ablowitz-Ladik chain (AL) are studied. It is shown that for $n$-sites the dynamical algebra of DNLS Hamilton operator is given by…
Classical Hamiltonian systems with balanced loss and gain are considered in this review. A generic Hamiltonian formulation for systems with space-dependent balanced loss and gain is discussed. It is shown that the loss-gain terms may be…
We analyze the stability of a non-Hermitian coupler with respect to modulational inhomogeneous perturbations in the presence of unbalanced gain and loss. At the parity-time (PT) symmetry point the coupler is unstable. Suitable symmetry…
The phenomenon of PT (parity- and time-reversal) symmetry breaking is conventionally associated with a change in the complex mode spectrum of a non-Hermitian system that marks a transition from a purely oscillatory to an exponentially…
Two alternative scenarios are shown possible in Quantum Mechanics working with non-Hermitian $PT-$symmetric form of observables. While, usually, people assume that $P$ is a self-adjoint indefinite metric in Hilbert space (and that their…
This paper proposes some efficient and accurate adaptive two-grid (ATG) finite element algorithms for linear and nonlinear partial differential equations (PDEs). The main idea of these algorithms is to utilize the solutions on the $k$-th…
We study the interplay between nonlinearity in exciton transport and trapping due to a sink site for the dimer and the trimer with chain configuration by a numerical integration of the discrete nonlinear Schroedinger equation. Our results…
We introduce a one-dimensional PT-symmetric system, which includes the cubic self-focusing, a double-well potential in the form of an infinitely deep potential box split in the middle by a delta-functional barrier of an effective height…
One of the challenges of the modern photonics is to develop all-optical devices enabling increased speed and energy efficiency for transmitting and processing information on an optical chip. It is believed that the recently suggested…
In an anti-Hermitian linear system, all energy eigenvalues are purely imaginary and the corresponding eigenvectors are orthogonal. This implies that no stationary state is available in such systems. We consider an anti-Hermitian lattice…
We address propagation of light in nonlinear twisted multi-core fibers with alternating amplifying and absorbing cores that are arranged into the PT - symmetric structure. In this structure, the coupling strength between neighboring cores…