Modulated wave patterns brought by higher-order dispersion and cubic–quintic nonlinearity in monoatomic chains with anharmonic potential.

This work investigates the modulation instability and wave patterns in the extended nonlinear Schrödinger equation with higher-order dispersion and cubic–quintic nonlinearity. We use the one-dimensional monoatomic chain with anharmonic potential to derive the discrete model. From the multiple scale method combined with a quasidiscreteness approximation, we derive the cubic–quintic nonlinear schrödinger equation, and thereafter an expression of the modulation instability gain is obtained by using a linearized expression. One notices that the modulation instability growth is sensitive to the higher-order dispersion and nonlinearities terms. A split-step Fourier method is used to assess the analytical predictions. A long evolution of the continuous wave is shown to lead to the formation of the bright soliton, and Akhemediev breathers also emerge to manifest the modulation instability. We have also demonstrated that the excitation wave number generates the train of waves to confirm the fact that the continuous wave can grow exponentially with any value of the latter. We mention equally that the model of the extended cubic–quintic nonlinearity with complex envelope has opened new features of the modulated wave patterns in monoatomic chains. • The MI and waves patterns of the NLSE are investigated. • To obtain CQ NLSE we use multiple scale method and quasi-discreteness approximation. • A split-step Fourier method is used to assess the analytical predictions. • Bright solitons and Akhemediev breathers also emerges to manifestthe MI. • The CW can grow exponentially with any value of the excitation wave number. [ABSTRACT FROM AUTHOR]