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We consider the asymptotic behavior of the solutions of a nonlinear Schrodinger (NLS) model incorporating linear and nonlinear gain/loss. First, we describe analytically the dynamical regimes (depending on the gain/loss strengths), for finite-time collapse, decay, and global existence of solutions in the dynamics. Then, for all the above parametric regimes, we use direct numerical simulations to study the dynamics corresponding to algebraically decaying initial data. We identify crucial differences between the dynamics of vanishing initial conditions, and those converging to a finite constant background: in the former (latter) case we find strong (weak) collapse or decay, when the gain/loss parameters are selected from the relevant regimes. One of our main results, is that in all the above regimes, non-vanishing initial data transition through spatiotemporal, algebraically decaying waveforms. While the system is nonintegrable, the evolution of these waveforms is reminiscent to the evolution of the Peregrine rogue wave of the integrable NLS limit. The parametric range of gain and loss for which this phenomenology persists is also touched upon.
We discuss the finite-time collapse, also referred as blow-up, of the solutions of a discrete nonlinear Schr{o}dinger (DNLS) equation incorporating linear and nonlinear gain and loss. This DNLS system appears in many inherently discrete physical cont
Following the concept of $mathcal{PT}$-symmetric couplers, we propose a linearly coupled system of nonlinear waveguides, made of positive- and negative-index materials, which carry, respectively, gain and loss. We report novel bi- and multi-stability
We perform a numerical study of the initial-boundary value problem, with vanishing boundary conditions, of a driven nonlinear Schrodinger equation (NLS) with linear damping and a Gaussian driver. We identify Peregrine-like rogue waveforms, excited by
We present study of the dynamics of two ring waveguide structure with space dependent coupling, linear gain and nonlinear absorption - the system that can be implemented in polariton condensates, optical waveguides, and nanocavities. We show that by
A Parity-Time (PT)-symmetric system with periodically varying-in-time gain and loss modeled by two coupled Schrodinger equations (dimer) is studied. It is shown that the problem can be reduced to a perturbed pendulum-like equation. This is done by fi