Plane wave solution of extended discrete nonlinear schrödinger equation

In this paper, we considered the extended discrete nonlinear Schrodinger equation (EDNLSE) which includes the nearest neighbour nonlinear interaction in addition to the on-site cubic and quintic nonlinearities. The objective of this study is to investigate the modulational instability of plane matter-wave solution in dipolar Bose-Einstein Condensates (BEC) in a periodic optical lattice and to compare the analytical results with numerical. Analytically, the problem is solved by using perturbed solution of the plane wave where the instability of the gain can be obtained. The conditions of the stability of the plane wave had been analysed and confirmed numerically, by applications of Runge-Kutta method. Three specific cases were studied where only cubic-quintic nonlinearity  is considered, only quintic-dipolar ( ) is considered and lastly non-zero for all terms. The numerical results are aligned with the analytical results.