Nonlinear Schrödinger solitons oscillate under a constant external force

Abstract

We investigate the dynamics of solitons of the cubic nonlinear Schrödinger equation with an external time-independent force of the form f(x)=rexp(−iKx). Here the solitons travel with an oscillating velocity and all other characteristics of the solitons (amplitude, width, momentum, and phase) also oscillate. This behavior was predicted by a collective variable theory and confirmed by simulations. However, the reason for these oscillations remains unclear. Moreover, the spectrum of the oscillations exhibits a second strong peak, in addition to the intrinsic soliton peak. We show that the additional frequency belongs to a certain extended linear mode (which we refer to as a phonon for short) close to the lower band edge of the phonon continuum. Initially the soliton is at rest. When it starts to move it is deformed, begins to oscillate, and excites the above phonon mode such that the total momentum in a certain moving frame is conserved. In this frame the phonon does not move. However, not only does the soliton move in the homogeneous, time-periodic field of the phonon, but it also oscillates.