Modulational instability and exact solutions of the discrete cubic–quintic Ginzburg–Landau equation

Abstract

In this paper, we investigate analytically and numerically the modulational instability in a model of nonlinear physical systems such as Bose–Einstein condensates in a deep optical lattice. This model is described by the discrete complex cubic–quintic Ginzburg–Landau equation with a non-local quintic term. We obtain characteristics of the modulational instability in the form of typical dependences of the instability growth rate (gain) on the perturbation wavenumber and the system's parameters. Excellent agreement has been obtained between the analytical and numerical study. Further, we derive the periodic function and new type of solitary wave solutions for the above system. By using the extended Jacobi elliptic function approach, we obtain the exact stationary solitons and periodic wave solutions of this equation. These solutions include the Jacobi periodic wave solution, alternating phase Jacobi periodic wave solution, kink and bubble soliton solutions, alternating phase kink soliton and alternating phase bubble soliton solutions.