Riemann-Hilbert approach for the Camassa-Holm equation on the line

We present a Riemann-Hilbert problem formalism for the initial value problem for the Camassa-Holm equation u(t) - u(txx) + 2 omega u(x) + 3uu(x) = 2u(x)u(xx) + uu(xxx) on the line (CH). We show that: (i) for all omega > 0, the solution of this problem can be obtained in a parametric form via the solution of some associated Riemann-Hilbert problem; (ii) for large time, it develops into a train of smooth solitons; (iii) for small omega, this soliton train is close to a train of peakons, which are piecewise smooth solutions of the CH equation for omega = 0.