Example of a Relativistic, Completely Integrable, Hamiltonian System

The initial value problem for a system of two relativistic scalar fields in two space-time dimensions is solved. One of them has a self-interaction of the sine-Gordon type, while the other is massless and moves in a background geometry which has a dynamical evolution of its own. The system has soliton solutions and is shown to be completely integrable using the inverse scattering method. The appropriate linear equations are found by thinking of the problem as the isometric embedding of a surface in a three-dimensional Euclidean space.