Solitons on pseudo-Riemannian manifolds I. The sine-Gordon equation

The sine-Gordon equation on R1+1 with background pseudo-Riemannian metric expressed in conformal co-ordinates as g = epsilon(-2)e(2 rho)(dt(2) - dx(2)) is studied. If rho is independent of t, x the equation admits a two parameter family of soliton solutions in which the soliton moves along straight time-like lines. In the scaling epsilon --> 0, here called the particle limit, the soliton has size epsilon. In this limit it is proved that for non-constant rho = rho(t, x) solutions exist which represent solitons concentrated along time-like geodesics of g. There is a close relation of the present problem to geometric optics, with the difference that it is concerned with describing energy concentration rather than oscillations.