Localized modes in inhomogeneous one-dimensional anharmonic lattices

We present numerical calculations for the determination of localized modes in one-dimensional finite chains of atoms with free ends containing harmonic and quartic anharmonic interactions. By adding step by step the quartic term we can follow the formation of even and odd localized modes arising from the highest harmonic frequency mode. We have studied the role of crystal inhomogeneity by introducing a modification of the fourth-order force constant between neighboring atoms at the center of the chain, where the localized mode has its maximum displacement. For large weakening of this force constant the localized mode develops a double-peaked structure, as has been found in the continuum limit. In the case of asymmetrical local inhomogeneity the localized mode remains stable and moves toward the atom with the inhomogeneity. We also show the existence of anharmonic surface modes localized at the end of the chain.