Internal modes of envelope solitons

We extend the concept of internal mode to envelope solitons and show that this mode is responsible for long-lived, weakly damped periodic oscillations of the soliton amplitude observed in numerical simulations. We present analytical and numerical results for solitons of the generalized nonlinear Schrödinger equation and analyze the example of the cubic-quintic nonlinearity in more detail. We obtain also analytical criteria for the existence and bifurcations of the soliton internal mode and calculate the rate of the radiation-induced damping of the soliton oscillations induced by excitation of the internal mode.