Sine-Gordon kinks on a discrete lattice. I. Hamiltonian formalism

We derive a complete Hamiltonian formalism for a kink on a one-dimensional discrete lattice in which the position of the center of the kink appears as one of the canonical variables. Our method is a generalization to the discrete lattice of the method used in field theory to introduce the soliton as a canonical degree of freedom. The derivation is valid for a particle chain in a periodic potential when there exists a solitary-wave solution in the continuum limit. We show that the discrete lattice is responsible for an adiabatic dressing of the kink and for spontaneous emission of phonons. In the limit where the effective length of the kink is much larger than the interparticle spacing the kink experiences the well-known periodic Peierls-Nabarro potential. In the case of a short kink, the discrete lattice causes the continuum kink configuration to be adiabatically dressed, leading to a renormalization of the Peierls-Nabarro potential and in turn to an enhancement of corresponding small-amplitude oscillatory frequency. In addition, we formally derive an equation that describes the radiation of phonons by the moving kink, an effect of the lattice discreteness.