Long-time Asymptotics of the One-dimensional Damped Nonlinear Klein-Gordon Equation

For the one-dimensional nonlinear damped Klein-Gordon equation partial derivative(2)(t) u + 2 alpha partial derivative(t)u - partial derivative(2)(x) u + u - |u|(p-1)u = 0 on R x R, with alpha > 0 and p > 2, we prove that any global finite energy solution either converges to 0 or behaves asymptotically as t -> infinity as the sum of K >= 1 decoupled solitary waves. In the multi-soliton case K >= 2, the solitary waves have alternate signs and their distances are of order log t.