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 prove finite time supercritical blowup in a parity-time-symmetric system of the two coupled nonlinear Schrodinger (NLS) equations. One of the equations contains gain and the other one contains dissipation such that strengths of the gain and dissipation are equal. We address two cases: in the first model all nonlinear coefficients (i.e. the ones describing self-action and non-linear coupling) correspond to attractive (focusing) nonlinearities, and in the second case the NLS equation with gain has attractive nonlinearity while the NLS equation with dissipation has repulsive (defocusing) nonlinearity and the nonlinear coupling is repulsive, as well. The proofs are based on the virial technique arguments. Several particular cases are also illustrated numerically.

http://arxiv.org/abs/1407.2438
Analysis of PDEs (math.AP); Optics (physics.optics)

We introduce a stochastic PT-symmetric coupler, which is based on dual-core waveguides with fluctuating parameters, such that the gain and the losses are exactly balanced in average. We consider different parametric regimes which correspond to the broken and unbroken PT symmetry, as well as to the exceptional point of the underlying deterministic system. We demonstrate that in all the cases the statistically averaged intensity of the field grows. This result holds for either linear or nonlinear coupler and is independent on the type of fluctuations.

http://arxiv.org/abs/1401.6352
Optics (physics.optics)

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 propose a scheme of creating a tunable highly nonlinear defect in a one-dimensional photonic crystal. The defect consists of an atomic cell filled in with two isotopes of three-level atoms. The probe-field refractive index of the defect can be made parity-time (PT) symmetric, which is achieved by proper combination of a control field and of Stark shifts induced by a far-off-resonance field. In the PT-symmetric system families of stable nonlinear defect modes can be formed by the probe field.

http://arxiv.org/abs/1309.2839

Optics (physics.optics); Pattern Formation and Solitons (nlin.PS)

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 situations when the underlying linear problem is characterized by the presence of degenerate eigenvalues or exceptional-point singularity. In each of the cases we construct formal expansions for small-amplitude nonlinear modes. We also report a class of nonlinearities allowing for a system to admit one or several integrals of motion, which turn out to be determined by the pseudo-Hermiticity of the nonlinearity.

http://arxiv.org/abs/1306.5286
Pattern Formation and Solitons (nlin.PS)

The combination of the interference with the amplification of modes in a waveguide with gain and losses can result in a giant amplification of the propagating beam, which propagates without distortion of its average amplitude. An increase of the gain-loss gradient by only a few times results in a magnification of the beam by a several orders of magnitude.

http://arxiv.org/abs/1207.1792
Optics (physics.optics)