The Modes In PT-symmetric System And Their Stability

Recently,the concept of parity time(PT)symmetry has been a subject of intense interest in the field of quantum mechanics,because there not only implies real eigenvalues,but also guarantees probability conservation.Due to the equivalence between quantum mechanical Schr?dinger equation and optical wave equation,the concept of PT symmetry was extended to classical optical systems with complex valued external potentials by use of optical amplification(gain)and absorption(loss).The waveguides with PT symmetric is simple,but can show some novel properties.For example,power oscillation,non-reciprocal light propagation,unidirectional invisibility and PT symmetric break.The properties of non-reciprocal optical wave propagation may be applied to optical devices designing,such as spatial optical switch,single-mode laser amplifiers,and optical beams controlling.Further,the concept of parity time symmetry is also extended in nonlinear optical waveguides which expands the studying of regular soliton and become a hot topic.In this paper,we focus on the studies of modes in PT-symmetric waveguides,including mulit-soliton,breather,constant-intensity solution and periodic solutions and analysis their dynamics,which involves three parts:1.The vector soliton solutions,including vector multisoliton solutions and the vector soliton solutions on finite background,for the coupled NLS equations with linear-coupling and gain-loss effects have been presented by employing the Darboux transformation.The results show that vector one-and two-soliton solutions,and even multisoliton solutions are stable in certain regions against both initial random perturbation for amplitude and longitudinal random fluctuation along propagation direction for gain-loss.We also construct a soliton chain with alternating signs of adjacent solitons to demonstrate the vector multisoliton Newton's cradle dynamics.Based on vector soliton solutions on a finite background,nonlinear Talbot recurrence effects excited by linearly modulated continuous waves are discussed,and the results show that the recurrence patterns of the nonlinear Talbot effects can be drawn by suitably choosing a frequency modulation factor.The former initial input states reemerge at the Talbot length and the half Talbot length with a half period shift,the latter only reemerge at the Talbot length.Finally,the evolution of the vector Peregrine solution is also studied by initially exciting a small localized perturbation on a continuous wave background,and the splitting feature can be exhibited,so we use continuous wave pumping and spectral filter can realize the optical amplifier and the long-distance transmission.2.A class of constant-amplitude(CA)solutions of the nonlinear Schr?dinger equation with the third-order spatial dispersion(TOD)and complex potentials are given by using inverse method.It is shown that the diffraction of truncated CA states with a correct phase structure can be strongly suppressed,and the results show that the TOD term tends to attenuate the Modulational Instability(MI).In particular,simulations demonstrate a phenomenon of weak stability,which occurs when the linear-stability analysis predicts small values of the MI growth rate.3.First,the mode and the stability in PT-symmetry waveguides with Scarff-II potential are considered.Next,we study families of periodic solutions in both nonlinear Schr?dinger equation and nonlinear Schr?dinger equation with periodically modulated third order dispersion(TOD).Without TOD,even if they are unstable against small random perturbations,the results show that the periodic solutions may be stable against perturbations in specific Floquet-Bloch bands,due to the existence of the respective stability band.When amplitude A is small,the periodic solutions are stable at all wavenumber k,and they also be stable against small random-noise perturbations,i.e.,dynamical stable.With periodically-modulated third-order dispersion,the results show that the band stability domains of the periodic solutions may be narrowed when increasing the nonlinear effects and the amplitudes of periodic solutions.