We present a Riemann-Hilbert problem formalism for the initial value problem for the Camassa-Holm equation ut-utxx + 2ωux + 3uux= 2uxuxx + uuxxx on the line (CH), where ω is a positive parameter. We show that, for all ω > 0, the solution of this initial value problem can be obtained in a parametric form from the solution of some associated Riemann-Hilbert problem; that for large time, it develops into a train of smooth solitons; and that 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

1. Introduction

The main purpose of this paper is to develop an inverse scattering approach, based on an appropriate Riemann–Hilbert problem formulation, for the initial value problem for the Camassa–Holm (CH) equation [Camassa and Holm 1993] on the line, whose form is where ω is a positive parameter. The CH equation is a model equation describing the shallow-water approximation in inviscid hydrodynamics. In this equation u = u(x, t) is a real-valued function that refers to the horizontal fluid velocity along the x direction (or equivalently, the height of the water's free surface above a flat bottom) as measured at time t . The constant ω is related to the critical shallow water wave speed, where g is the acceleration of gravity and h0 is the undisturbed water depth; hence, the case ω >0 is physically more relevant than the case ω = 0, though the latter has attracted more attention in the mathematical studies due to interesting specific features such as the existence of peaked (nonanalytic) solitons.