Dynamics of the Toda lattice: A soliton-phonon phase-shift analysis

Abstract

The dynamics of the Toda lattice can be described in terms of solitons and nonlinear phonons (ripples). The latter are shown to be obtainable from a general multisoliton solution by allowing the soliton parameters to become imaginary. This device yields phonon phase shifts due to a soliton via a linearization procedure and shows that the zone-edge phonon (standing wave) is removed in the presence of a soliton. An expansion of the soliton-ripple solution up to the second order in the phonon amplitude allows the calculation of typical nonlinear quantities such as momentum and mass of the ripple and the spatial shift of a soliton due to its collision with a ripple. Finally, the relationship is established between the form of a ripple, viewed as a phonon wave packet, and its associated action variable within the framework of inverse scattering theory.