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 t x x + 2 ω u x + 3 u u x = 2 u x u x x + u u x x x on the line (CH). We show that: (i) for all ω > 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 ω , this soliton train is close to a train of peakons, which are piecewise smooth solutions of the CH equation for ω = 0 .