Bright and peaklike pulse solitary waves and analogy with modulational instability in an extended nonlinear Schr¨odinger equation

摘要

The modulational instability (MI) phenomenon in the nonlinear Schr¨odinger equation (NLSE) extended byntwo different nonlinear dispersion terms and the gradient term is investigated. We find that the possibility ofninstability of plane waves depends on the sign of the nonlinear dispersion parameters with regard to the linearndispersion coefficient. In contrast to the basic NLSE, the system may exhibit instability in the defocusing medianfor amplitude exceeding a critical value depending on the magnitude of the nonlinear dispersion. An additionalnfeature, namely the higher order or the infinite gain band, absent in the NLSE case, may appear and in whichnMI induces the birth of the nonlinear localized wave (NLW) of different carrier wave numbers. The result ofnthe qualitative investigations of the system’s dynamics indicates the existence of the NLW, such as peak, bright,ndark, and compact dark solitary waves which can be well predicted by the MI criteria. In addition the nonlinearndispersion induces the existence of a pair of bright-dark solitary waves which is usually exhibited by the couplednNLSEs only, and the pairs of peak-dark and compact dark-bright solitary waves.