Bright and dark small amplitude nonlinear localized modes in a quantum one-dimensional Klein--Gordon chain

By means of the Glauber's coherent state method combined with multiple-scale method, this paper investigates the localized modes in a quantum one-dimensional Klein--Gordon chain and finds that the equation of motion of annihilation operator is reduced to the nonlinear Schrödinger equation. Interestingly, the model can support both bright and dark small amplitude travelling and non-travelling nonlinear localized modes in different parameter spaces.