Parts II, III and IV of this series are devoted to proving long time stability of solitary waves in one-dimensional nonintegrable lattices with Hamiltonian with a general nearest-neighbour potential V. Here in part III we analyse the evolution equation obtained by linearizing the dynamics at a solitary wave. This equation is nonautonomous, because discrete solitary waves are not time-independent modulo a spatial shift (like their continuous counterparts), but time-periodic modulo a spatial shift.

We develop a Floquet theory modulo shifts on the lattice that naturally characterizes the time-t evolution on the lattice in terms of a strongly continuous group of operators on the real line, in a manner reminiscent of Howland's treatment of quantum scattering with time-periodic potentials. This allows us to reduce the main hypothesis of our nonlinear stability theorem in part II (namely, exponential decay in the linearized dynamics on the symplectic complement to the solitary-wave manifold) to an eigenvalue condition on the generator of the group, which is a differential-difference operator on the real line. Physically, the eigenvalue condition means that no spatially localized modes of constant shape exist which travel at the solitary wave speed and have exponentially growing or neutral amplitude.