We study a system of two coupled nonlinear Schrodinger equations, where one equation includes gain and the other one includes losses. Strengths of the gain and the loss are equal, i.e., the resulting system is parity-time (PT) symmetric. The model includes both linear and nonlinear couplings, such that when all nonlinear coefficients are equal, the system represents the PT-generalization of the Manakov model. In the one-dimensional case, we prove the existence of a global solution to the Cauchy problem in energy space \(H_1\), such that the \(H_1\)-norm of the global solution may grow in time. In the Manakov case, we show analytically that the \(L_2\)-norm of the global solution is bounded for all times and numerically that the \(H_1\)-norm is also bounded. In the two-dimensional case, we obtain a constraint on the \(L_2\)-norm of the initial data that ensures the existence of a global solution in the energy space \(H_1\).

http://arxiv.org/abs/1411.2895
Analysis of PDEs (math.AP)

We generalize a finite parity-time (PT-)symmetric network of the discrete nonlinear Schrodinger type and obtain general results on linear stability of the zero equilibrium, on the nonlinear dynamics of the dimer model, as well as on the existence and stability of large-amplitude stationary nonlinear modes. A result of particular importance and novelty is the classification of all possible stationary modes in the limit of large amplitudes. We also discover a new integrable configuration of a PT-symmetric dimer.

http://arxiv.org/abs/1310.5651
Pattern Formation and Solitons (nlin.PS); Dynamical Systems (math.DS)

We consider nonlinear dynamics in a finite parity-time-symmetric chain of the discrete nonlinear Schrodinger (dNLS) type. We work in the range of the gain and loss coefficient when the zero equilibrium state is neutrally stable. We prove that the solutions of the dNLS equation do not blow up in a finite time and the trajectories starting with small initial data remain bounded for all times. Nevertheless, for arbitrary values of the gain and loss parameter, there exist trajectories starting with large initial data that grow exponentially fast for larger times with a rate that is rigorously identified. Numerical computations illustrate these analytical results for dimers and quadrimers.

http://arxiv.org/abs/1307.2973
Exactly Solvable and Integrable Systems (nlin.SI); Mathematical Physics (math-ph)

In the present work we examine both the linear and nonlinear properties of two related 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 at an analogue of the anti-continuum limit for the dNLS equation. Secondly, we consider the case when a finite PT-dNLS chain is embedded as a defect in the infinite dNLS lattice. We show that the stability intervals of the infinite PT-dNLS lattice are wider than in the case of a finite PT-dNLS chain. We also prove existence of localized stationary states (discrete solitons) in the analogue of the anti-continuum limit for the dNLS equation.
Numerical computations illustrate the existence of nonlinear stationary states, as well as the stability and saddle-center bifurcations of discrete solitons.

http://arxiv.org/abs/1303.3298
Pattern Formation and Solitons (nlin.PS); Dynamical Systems (math.DS)

We illustrate, through a series of prototypical examples, that linear parity-time (PT) symmetric lattices with extended gain/loss profiles are generically unstable, for any non-zero value of the gain/loss coefficient. Our examples include a parabolic real potential with a linear imaginary part and the cases of no real and constant or linear imaginary potentials. On the other hand, this instability can be avoided and the spectrum can be real for localized or compact PT-symmetric potentials. The linear lattices are analyzed through discrete Fourier transform techniques complemented by numerical computations.

http://arxiv.org/abs/1211.5815
Exactly Solvable and Integrable Systems (nlin.SI); Mathematical Physics (math-ph); Quantum Physics (quant-ph)