Impurity Localized Modes of Nonlinear Lattice Systems

Impurity localized modes of nonlinear lattice systems are studied by applying the method of multiple scales. The solutions of the localized modes of both infinite and semi-infinite nonlinear lattice systems are obtained up to the third order of the amplitude. According to the sign of the parameter specifying the second order nonlinearity, nonlinear lattice systems expand or shrink in the vicinity of impurity. Moreover, the frequencies of the localized modes are shown to decrease or increase with their amplitude, depending on the magnitude of the parameters specifying the second and the third order nonlinearities. As an example, numerical experiments for the Toda lattice with an impurity and the semi-infinite Toda lattice with an impurity are performed, and the results are in good agreement with those obtained by the perturbation method.