Stability and instability of nonlinear defect states in the coupled mode equations -- analytical and numerical study

摘要

Coupled backward and forward wave amplitudes of an electromagnetic fieldpropagating in a periodic and nonlinear medium at Bragg resonance are governedby the nonlinear coupled mode equations (NLCME). This system of PDEs, similarin structure to the Dirac equations, has gap soliton solutions that travel atany speed between 0 and the speed of light. A recently considered strategy forspatial trapping or capture of gap optical soliton light pulses is based on theappropriate design of localized defects in the periodic structure. Localizeddefects in the periodic structure give rise to defect modes, which persist as{\it nonlinear defect modes} as the amplitude is increased. Soliton trapping isthe transfer of incoming soliton energy to {\it nonlinear} defect modes. Toserve as targets for such energy transfer, nonlinear defect modes must bestable. We therefore investigate the stability of nonlinear defect modes.Resonance among discrete localized modes and radiation modes plays a role inthe mechanism for stability and instability, in a manner analogous to thenonlinear Schr\"odinger / Gross-Pitaevskii (NLS/GP) equation. However, thenature of instabilities and how energy is exchanged among modes is considerablymore complicated than for NLS/GP due, in part, to a continuous spectrum ofradiation modes which is unbounded above and below. In this paper we (a)establish the instability of branches of nonlinear defect states which, forvanishing amplitude, have a linearization with eigenvalue embedded within thecontinuous spectrum, (b) numerically compute, using Evans function, thelinearized spectrum of nonlinear defect states of an interesting multiparameterfamily of defects, and (c) perform direct time-dependent numerical simulationsin which we observe the exchange of energy among discrete and continuum modes.