Dmitry E. Pelinovsky, Dmitry A. Zezyulin, Vladimir V. Konotop
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)
João-Paulo Dias, Mário Figueira, Vladimir V. Konotop, Dmitry A. Zezyulin
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)
Yuli V. Bludov, Chao Hang, Guoxiang Huang, Vladimir V. Konotop
We study interaction of a soliton in a parity-time (PT) symmetric coupler which has local perturbation of the coupling constant. Such a defect does not change the PT-symmetry of the system, but locally can achieve the exceptional point. We found that the symmetric solitons after interaction with the defect either transform into breathers or blow up. The dynamics of anti-symmetric solitons is more complex, showing domains of successive broadening of the beam and of the beam splitting in two outwards propagating solitons, in addition to the single breather generation and blow up. All the effects are preserved when the coupling strength in the center of the defect deviates from the exceptional point. If the coupling is strong enough the only observable outcome of the soliton-defect interaction is the generation of the breather.
http://arxiv.org/abs/1405.1829
Optics (physics.optics)
Vladimir V. Konotop, Dmitry A. Zezyulin
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)
Yaroslav V. Kartashov, Vladimir V. Konotop, Fatkhulla Kh. Abdullaev
We report a diversity of stable gap solitons in a spin-orbit coupled Bose-Einstein condensate subject to a spatially periodic Zeeman field. It is shown that the solitons, can be classified by the main physical symmetries they obey, i.e. symmetries with respect to parity (P), time (T), and internal degree of freedom, i.e. spin, (C) inversions. The conventional gap and gap-stripe solitons are obtained in lattices with different parameters. It is shown that solitons of the same type but obeying different symmetries can exist in the same lattice at different spatial locations. PT and CPT symmetric solitons have anti-ferromagnetic structure and are characterized respectively by nonzero and zero total magnetizations.
http://arxiv.org/abs/1310.8517
Quantum Gases (cond-mat.quant-gas); Pattern Formation and Solitons (nlin.PS)
Dmitry E. Pelinovsky, Dmitry A. Zezyulin, Vladimir V. Konotop
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)
A. V. Yulin, V. V. Konotop
Stable discrete compactons in arrays of inter-connected three-line waveguide arrays are found in linear and nonlinear limits in conservative and in parity-time PT symmetric models. The compactons result from the interference of the fields in the two lines of waveguides ensuring that the third (middle) line caries no energy. PT-symmetric compactons require not only the presence of gain and losses in the two lines of the waveguides but also complex coupling, i.e. gain and losses in the coupling between the lines carrying the energy and the third line with zero field. The obtained compactons can be stable and their branches can cross the branches of the dissipative solitons. Unusual bifurcations of branches of solitons from linear compactons are described.
http://arxiv.org/abs/1310.5328
Optics (physics.optics); Pattern Formation and Solitons (nlin.PS)
Chao Hang, Dmitry A. Zezyulin, Vladimir V. Konotop, Guoxiang Huang
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)
Dmitry A. Zezyulin, Vladimir V. Konotop
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)
Yu.V. Bludov, R. Driben, V.V. Konotop, B.A. Malomed
We considered the modulational instability of continuous-wave backgrounds, and the related generation and evolution of deterministic rogue waves in the recently introduced parity-time (PT)-symmetric system of linearly-coupled nonlinear Schr\”odinger equations, which describes a Kerr-nonlinear optical coupler with mutually balanced gain and loss in its cores. Besides the linear coupling, the overlapping cores are coupled through cross-phase-modulation term too. While the rogue waves, built according to the pattern of the Peregrine soliton, are (quite naturally) unstable, we demonstrate that the focusing cross-phase-modulation interaction results in their partial stabilization. For PT-symmetric and antisymmetric bright solitons, the stability region is found too, in an exact analytical form, and verified by means of direct simulations.
http://arxiv.org/abs/1304.7369
Optics (physics.optics); Pattern Formation and Solitons (nlin.PS)