Exponential decay of H 1 -localized solutions and stability of the train of N solitary waves for the Camassa–Holm equation

For the Camassa-Holm equation with kappa> or =0, we first prove that any global solution that is H1-localized and moves fast enough to the right decays exponentially in space uniformly with respect to time. We also prove that for kappa>0, a train of N solitary waves, which are sufficiently decoupled, is orbitally stable in H1(R).