The structure of eigenmodes and phonon scattering by discrete breathers in the discrete nonlinear Schrِdinger chain

We present a linear theory for one-dimensional phonon scattering by discrete breathers in the discrete nonlinear Schrödinger equation using the transfer matrix formulation. We focus on eigenmodes in the linearized equation, which plays an important role in the scattering problem. Considering a special class of boundary conditions for both physical and unphysical eigenmodes in the non-traveling region and their continuation into the traveling region, we obtain an intuitive picture of the relation between the occurrence of perfect transmission and the localized eigenmode threshold. The perturbation approach with a transfer matrix formulation in the weak coupling limit predicts both the existence of two localized eigenmode thresholds at finite coupling strength and the structure of perfect transmission and perfect reflection. These results are shown to be applicable to a wide class of nonlinear chains including the phonon scattering problem by the discrete breather in the Klein–Gordon chain with cubic on-site potential.