Related papers: Nonlinear dynamics in PT-symmetric lattices
We consider a class of nonlinear Schr\"odinger equation in two space dimensions with an attractive potential. The nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical (in $L^2$)…
We investigate the dissipative dynamics of linear and nonlinear waves in harmonic traps by means of engineered complex non-Hermitian potentials. By combining an analytical mapping between real and complex Schr\"odinger equations with direct…
Using spatial domain techniques developed by the authors and Myunghyun Oh in the context of parabolic conservation laws, we establish under a natural set of spectral stability conditions nonlinear asymptotic stability with decay at Gaussian…
We introduce a model based on a system of coupled nonlinear Schrodinger (NLS) equations with opposite signs infront of the kinetic and gradient terms in the two equations. It also includes time-dependent nonlinearity coefficients and a…
In this work we study a system of Schr\"odinger equations involving nonlinearities with quadratic growth. We establish sharp criterion concerned with the dichotomy global existence versus blow-up in finite time. Such a criterion is given in…
In the present paper we study the following scaled nonlinear Schr\"odinger equation (NLS) in one space dimension: \[ i\frac{d}{dt} \psi^{\varepsilon}(t) =-\Delta\psi^{\varepsilon}(t) +…
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…
Nonlinear dynamics of wave packets in two-dimensional parity-time-symmetric optical lattices near the phase-transition point are analytically studied. A novel fourth-order equation is derived for the envelope of these wave packets. A…
With perfectly balanced gain and loss, dynamical systems with indefinite damping can obey the exact PT-symmetry being marginally stable with a pure imaginary spectrum. At an exceptional point where the symmetry is spontaneously broken, the…
Lattice models with non-hermitian, parity and time-reversal ($\mathcal{PT}$) symmetric Hamiltonians, realized most readily in coupled optical systems, have been intensely studied in the past few years. A $\mathcal{PT}$-symmetric dimer…
We present what we believe to be the first known example of an exact quasiperiodic localized stable solution with spatially symmetric large-amplitude oscillations in a non-integrable Hamiltonian lattice model. The model is a one-dimensional…
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…
We study the evolution of a wave packet in a nonlinear Schr\"odinger lattice equation subject to a dc bias. In the absence of nonlinearity all normal modes are spatially localized giving rise to a Stark ladder with an equidistant eigenvalue…
We consider nonlinear modes of the nonlinear Schr\"odinger equation with a nonlocal nonlinearity and an additional PT-symmetric parabolic potential. We show that there exists a set of continuous families of nonlinear modes and study their…
We study the long time dynamics of the defocussing NLS equation. Compared with previous literature, we revisit the direct and inverse scattering map to obtain asymptotics in some weighted energy space that requires less restrictive decay…
We consider solutions of the defocusing nonlinear Schr\"odinger (NLS) equation on the half-line whose Dirichlet and Neumann boundary values become periodic for sufficiently large $t$. We prove a theorem which, modulo certain assumptions,…
Exact solutions to a nonlinear Schr{\"o}dinger lattice with a saturable nonlinearity are reported. For finite lattices we find two different standing-wave-like solutions, and for an infinite lattice we find a localized soliton-like…
We consider damped and forced discrete nonlinear Schr\"odinger equations on the lattice $\mathbb{Z}$. First we establish the existence of periodic and quasiperiodic breather solutions for periodic and quasiperiodic driving, respectively.…
Linear-threshold networks (LTNs) capture the mesoscale behavior of interacting populations of neurons and are of particular interest to control theorists due to their dynamical richness and relative ease of analysis. The aim of this paper…
In this paper, we analyze the long-time dynamics of small solutions to the $1d$ cubic nonlinear Schr\"odinger equation (NLS) with a trapping potential. We show that every small solution will decompose into a small solitary wave and a…