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)
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)
K. Li, D. A. Zezyulin, V. V. Konotop, P. G. Kevrekidis
In this work, we propose a PT-symmetric coupler whose arms are birefringent waveguides as a realistic physical model which leads to a so-called quadrimer i.e., a four complex field setting. We seek stationary solutions of the resulting linear and nonlinear model, identifying its linear point of PT symmetry breaking and examining the corresponding nonlinear solutions that persist up to this point, as well as, so-called, ghost states that bifurcate from them. We obtain the relevant symmetry breaking bifurcations and numerically follow the associated dynamics which give rise to growth/decay even within the PT-symmetric phase. Our obtained stationary nonlinear solutions are found to terminate in saddle-center bifurcations which are analogous to the linear PT-phase transition.
http://arxiv.org/abs/1212.1676
Quantum Physics (quant-ph)
D. A. Zezyulin, V. V. Konotop
By rearrangements of waveguide arrays with gain and losses one can simulate transformations among parity-time (PT-) symmetric systems not affecting their pure real linear spectra. Subject to such transformations, however, the nonlinear properties of the systems undergo significant changes. On an example of an array of four waveguides described by the discrete nonlinear Schr\”odinger equation with dissipation and gain, we show that the equivalence of the underlying linear spectra implies similarity of neither structure nor stability of the nonlinear modes in the arrays. Even the existence of one-parametric families of nonlinear modes is not guaranteed by the PT symmetry of a newly obtained system. Neither the stability is directly related to the PT symmetry: stable nonlinear modes exist even when the spectrum of the linear array is not purely real. We use graph representation of PT-symmetric networks allowing for simple illustration of linearly equivalent networks and indicating on their possible experimental design.
http://arxiv.org/abs/1202.3652
Pattern Formation and Solitons (nlin.PS); Optics (physics.optics)
D. A. Zezyulin, V. V. Konotop
By similarity transformations a parity-time (PT-) symmetric Hamiltonian can be reduced to a Hermitian or to another PT-symmetric Hamiltonian having the same linear spectrum. On an example of a PT-symmetric quadrimer we show that the spectral equivalence of different PT-symmetric and Hermitian systems implies neither similarity of the nonlinear modes nor their stability properties. Even the existence of one-parametric families of nonlinear modes is not guaranteed by the PT symmetry of the underlying linear system. Neither the stability is directly related to the PT symmetry: stable nonlinear modes exist beyond the PT symmetry breaking threshold. A “phase diagram” of a general PT-symmetric quadrimer allows for existence of “triple” points, where three different phases meet. We use graph representation of PT-symmetric networks giving simple illustration of linearly equivalent networks and indicating on their possible experimental design.
http://arxiv.org/abs/1202.3652
Pattern Formation and Solitons (nlin.PS); Optics (physics.optics)
D. A. Zezyulin, Y. V. Kartashov, V. V. Konotop
We report on detailed investigation of the stability of localized modes in the nonlinear Schrodinger equations with a nonlinear parity-time (alias PT) symmetric potential. We are particularly focusing on the case where the spatially-dependent nonlinearity is purely imaginary. We compute the Evans function of the linear operator determining the linear stability of localized modes. Results of the Evans function analysis predict that for sufficiently small dissipation localized modes become stable when the propagation constant exceeds certain threshold value. This is the case for periodic and \(\tanh\)-shaped complex potentials where the modes having widths comparable with or smaller than the characteristic width of the complex potential are stable, while broad modes are unstable. In contrast, in complex potentials that change linearly with transverse coordinate all modes are stable, what suggests that the relation between width of the modes and spatial size of the complex potential define the stability in the general case. These results were confirmed using the direct propagation of the solutions for the mentioned examples.
http://arxiv.org/abs/1111.0898
Pattern Formation and Solitons (nlin.PS); Mathematical Physics (math-ph); Optics (physics.optics)