Modulational instability and discrete breathers in the discrete cubic–quintic nonlinear Schrödinger equation

We investigate the properties of modulational instability and discrete breathers in the cubic–quintic discrete nonlinear Schrödinger equation. We analyze the regions of modulational instabilities of nonlinear plane waves. Using the Page approach [J.B. Page, Phys. Rev. B 41 (1990) 7835], we derive the conditions for the existence and stability for bright discrete breather solutions. It is shown that the quintic nonlinearity brings qualitatively new conditions for stability of strongly localized modes. The application to the existence of localized modes in the Bose–Einstein condensate (BEC) with three-body interactions in an optical lattice is discussed. The numerical simulations agree with the analytical predictions.