Intrinsic localized modes in the bulk and at the surface of anharmonic chains

In this paper we will review recents results relative to localized modes induced by anharmonicity in one-dimensional lattices. We will show that localized modes exists in monoatomic chains with and without a local inhomogeneity in the anharmonic force field. We will compare the discrete and the quasi-continuum intrinsic even and odd localized solutions. This analysis is carried out by taking into account harmonic and quartic anharmonic interactions. We will also present results for the diatomic chains showing the presence of surface modes and gap modes related to the maximum of the acoustic band. In this analysis will be also studied the effect of the cubic anharmonicity. One of the major effects of the cubic anharmonicity is to produce gap modes that split off from the bottom of the optical branch.