Quantum breathers and intrinsic localized excitation associated with the modulational instability in 1D Bose-Hubbard chain

In this work, the energy spectrum and intrinsic localized modes associated with the modulational instability of boson chains is described by a Bose-Hubbard model. By using of number states method combined with the numerical diagonalization, we show that the energy spectrum of the system exhibits bound states signature. With the help of multiple scales method in addition to a quasidiscreteness approximation, we obtain bright and dark type intrinsic localized modes at the center and at the edges of the Brillouin zone respectively and their appearance conditions. We found that the inclusion of interaction of two bosons at the same site can influence the appearance conditions of bright and dark soliton solutions. Considering the fact that nowadays, the forthright way to predict the formation of intrinsic localized modes in nonlinear systems is the modulational instability. We investigate analytically, through the linear stability of plane wave solutions, the existence of localized structures in the Bose-Hubbard chain. It is found that the shape of the region of modulational instability and the instability growth rates of the plane wave can be more affected dramatically when the interaction of two bosons at the same site is involved in the system. Furthermore, we calculate the instability growth rates of plane wave at the center and at the edge of the Brillouin zone for different values of the interaction of two bosons to analyze their formation conditions which are in full agreement with the method of multiple scale in addition to a quasi-discreteness approximation analysis.