Related papers: Nonlinear stationary states in PT-symmetric lattic…
We consider nonlinear dynamics in a finite parity-time-symmetric chain of the discrete nonlinear Schr{\"o}dinger (dNLS) type. We work in the range of the gain and loss coefficient when the zero equilibrium state is neutrally stable. We…
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 of the linear operator and consider a…
Dynamics of a chain of interacting parity-time invariant nonlinear dimers is investigated. A dimer is built as a pair of coupled elements with equal gain and loss. A relation between stationary soliton solutions of the model and solitons of…
We study a discrete nonlinear Schr\"odinger lattice with a parabolic trapping potential. The model, describing, e.g., an array of repulsive Bose-Einstein condensate droplets confined in the wells of an optical lattice, is analytically and…
Discrete solitons of the discrete nonlinear Schr\"odinger (dNLS) equation become compactly supported in the anti-continuum limit of the zero coupling between lattice sites. Eigenvalues of the linearization of the dNLS equation at the…
We investigate the dynamical behavior of continuous and discrete Schrodinger systems exhibiting parity-time (PT) invariant nonlinearities. We show that such equations behave in a fundamentally different fashion than their nonlinear…
We generalize a finite parity-time (${\cal PT}$-) symmetric network of the discrete nonlinear Schr\"odinger type and obtain general results on linear stability of the zero equilibrium, on the nonlinear dynamics of the dimer model, as well…
We consider the discrete solitons bifurcating from the anti-continuum limit of the discrete nonlinear Schr\"{o}dinger (NLS) lattice. The discrete soliton in the anti-continuum limit represents an arbitrary finite superposition of {\em…
We investigate branched PT-symmetric optical lattices. We consider both the linear and nonlinear Schr\"odinger equations with a PT-symmetric periodic potential on the graph and solve them by imposing weighted vertex boundary conditions. A…
We study the existence and stability of localized states in the discrete nonlinear Schr{\"o}dinger equation (DNLS) on two-dimensional non-square lattices. The model includes both the nearest-neighbor and long-range interactions. For the…
An overview is given of basic models combining discreteness in their linear parts (i.e. the models are built as dynamical lattices) and nonlinearity acting at sites of the lattices or between the sites. The considered systems include the…
In this paper we give a simple and short proof of asymptotic stability of soliton for discrete nonlinear Schr\"odinger equation near anti-continuous limit. Our novel insight is that the analysis of linearized operator, usually…
We derive a class of discrete nonlinear Schr{\"o}dinger (DNLS) equations for general polynomial nonlinearity whose stationary solutions can be found from a reduced two-point algebraic problem. It is demonstrated that the derived class of…
We obtain new results on the stability of discrete dark solitons bifurcating from the anti-continuum limit of the discrete nonlinear Schrodinger equation, following the analysis of our previous paper [Physica D 212, 1-19 (2005)]. We derive…
We study discrete vortices in the anti-continuum limit of the discrete two-dimensional nonlinear Schr{\"o}dinger (NLS) equations. The discrete vortices in the anti-continuum limit represent a finite set of excited nodes on a closed discrete…
We describe a mechanism that results in the nonlinear instability of stationary states even in the case where the stationary states are linearly stable. This instability is due to the nonlinearity-induced coupling of the linearization's…
We address the existence and stability of localized modes in the two-dimensional (2D) linear Schroedinger lattice with two symmetric nonlinear sites embedded into it, and a generalization for moderately localized nonlinearity featuring two…
We study the existence and stability of fundamental bright discrete solitons in a parity-time (PT)-symmetric coupler composed by a chain of dimers, that is modelled by linearly coupled discrete nonlinear Schrodinger equations with gain and…
We prove existence of discrete solitons in infinite parity-time (PT-) symmetric lattices by means of analytical continuation from the anticontinuum limit. The energy balance between dissipation and gain implies that in the anticontinuum…
The focusing nonlinear Schrodinger equation possesses special non-dispersive solitary type solutions, solitons. Under certain spectral assumptions we show existence and asymptotic stability of solutions with the asymptoic profile (as time…