On the Dynamics of the Classical Transverse Field Ising Chain

We investigate stationary and travelling wave solutions of a special lattice differential equation in one space dimension. Depending on a parameter λ, results are given on the existence, shape and stability for these kind of solutions. The analysis of travelling wave solutions leads us to a functional differential equation with both forward and backward shifts. The existence of solutions of this equation will be proved by use of the implicit function theorem. In particular, we consider kink solutions and periodic solutions