Nonlinear Stationary States in PT-Symmetric Lattices

  title={Nonlinear Stationary States in PT-Symmetric Lattices},
  author={Panayotis G. Kevrekidis and Dmitry E. Pelinovsky and Dmitry Tyugin},
  journal={SIAM J. Appl. Dyn. Syst.},
In the present work we examine both the linear and nonlinear properties of two related parity-time (PT)-symmetric systems of the discrete nonlinear Schrodinger (dNLS) type. First, we examine the parameter range for which the finite PT-dNLS chains have real eigenvalues and PT-symmetric linear eigenstates. We develop a systematic way of analyzing the nonlinear stationary states with the implicit function theorem. Second, we consider the case when a finite PT-dNLS chain is embedded as a defect in… 

Figures from this paper

Stationary modes and integrals of motion in nonlinear lattices with PT-symmetric linear part
We consider finite-dimensional nonlinear systems with linear part described by a parity-time (PT-) symmetric operator. We investigate bifurcations of stationary nonlinear modes from the eigenstates
Nonlinear dynamics in PT-symmetric lattices
We consider nonlinear dynamics in a finite parity-time-symmetric chain of the discrete nonlinear Schrodinger (dNLS) type. For arbitrary values of the gain and loss parameter, we prove that the
Bifurcation of nonlinear bound states in the periodic Gross-Pitaevskii equation with 𝒫𝒯-symmetry
Abstract The 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
Nonlinear modes and integrals of motion in finite PT-symmetric systems
We investigate bifurcations of nonlinear modes in parity-time (PT-) symmetric discrete systems. We consider a general class of nonlinearities allowing for existence of the nonlinear modes and address
Nonlinear modes in a generalized -symmetric discrete nonlinear Schrödinger equation
We generalize a finite parity-time (PT)-symmetric network of the discrete nonlinear Schr¨ odinger type and obtain general results on linear stability of the zero equilibrium, on the nonlinear
Discrete solitons and vortices on two-dimensional lattices of PT-symmetric couplers.
This work introduces a 2D network built of PT-symmetric dimers with on-site cubic nonlinearity, the gain and loss elements of the dimers being linked by parallel square-shaped lattices, and finds that unstable OnVs do not blow up, but spontaneously rebuild themselves into stable FSs.
Krein Signature in Hamiltonian and PT -Symmetric Systems
We explain the concept of Krein signature in Hamiltonian and PT symmetric systems on the case study of the one-dimensional Gross–Pitaevskii equation with a real harmonic potential and an imaginary


Eigenstates and instabilities of chains with embedded defects.
This work considers the eigenvalue problem for one-dimensional linear Schrödinger lattices with an embedded few-sites linear or nonlinear, Hamiltonian or non-conservative defect (an oligomer), and describes a general approach based on a matching of solutions of the linear portions of the lattice at the location of the oligomers.
Stability analysis for solitons in PT-symmetric optical lattices
Stability of solitons in parity-time (PT)-symmetric periodic potentials (optical lattices) is analyzed in both one- and two-dimensional systems. First we show analytically that when the strength of
Asymmetric wave propagation through nonlinear PT-symmetric oligomers
In the present paper, we consider nonlinear PT-symmetric dimers and trimers (more generally, oligomers) embedded within a linear Schrodinger lattice. We examine the stationary states of such chains
Nonlinear modes in finite-dimensional PT-symmetric systems.
It is shown that the equivalence of the underlying linear spectra does not imply similarity of the structure or stability of the nonlinear modes in the arrays, and a graph representation of PT-symmetric networks is used allowing for a simple illustration of linearly equivalent networks and indicating their possible experimental design.
Nonlinear PT-symmetric plaquettes
We introduce four basic two-dimensional (2D) plaquette configurations with onsite cubic nonlinearities, which may be used as building blocks for 2D -symmetric lattices. For each configuration, we
PT-symmetric oligomers: analytical solutions, linear stability, and nonlinear dynamics.
  • K. Li, P. Kevrekidis
  • Mathematics
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2011
This work focuses on the case of (few-site) configurations respecting the parity-time (PT) symmetry, i.e., with a spatially odd gain-loss profile, with nontrivial properties in their linear stability and in their nonlinear dynamics.
Solitons in a chain of parity-time-invariant dimers.
A relation between stationary soliton solutions of the model and solitons of the discrete nonlinear Schrödinger (DNLS) equation is demonstrated, and approximate solutions forsolitons whose width is large in comparison to the lattice spacing are derived, using a continuum counterpart ofThe discrete equations.
Exponentially fragile PT symmetry in lattices with localized eigenmodes.
The effect of localized modes in lattices of size N with parity-time (PT) symmetry, arranged in pairs of quasidegenerate levels with splitting delta, shows a cascade of bifurcations during which a pair of real levels becomes complex.
Solitons in PT-symmetric nonlinear lattices
The existence of localized modes supported by the $\mathcal{PT}$-symmetric nonlinear lattices is reported. The system considered reveals unusual properties: unlike other typical dissipative systems,