Aspects of exact dynamics for general solutions of the sine-Gordon equation with applications to domain walls

A knowledge of the dynamics of nonlinear systems with multiple excitations is important for calculation of the continuum density of states and for one method of calculation of the classical statistical mechanics of the sine-Gordon Hamiltonian. Hirota's $N$-soliton construction for the sine-Gordon equation is extended by analytic continuation to incorporate usefully breather and continuum solutions as well. The significance of these classical solutions is illustrated in the context of the propagation and excitations of domain walls in a classical uniaxial ferromagnet of infinite extent. From these solutions, formulas for the relative phase shifts in the scattering of a soliton, breather, or continuum solution from any one of these, acting as scatterer, are derived. Following Hirota, it is shown that under certain assumptions about the asymptotic properties of the solution the relative phase shifts of solutions corresponding to many soliton, breather, and continuum states may be found by adding the derived pairwise phase shifts.