The short-wave model for the Camassa–Holm equation: a Riemann–Hilbert approach

Abstract

We present the inverse scattering transform approach to the Cauchy problem on the line for the short-wave model for the Camassa–Holm equation in the form of an associated Riemann–Hilbert problem. This approach allows us to give a representation of the classical (smooth) solutions, describe their asymptotics as t → ∞ and describe cuspons—non-smooth soliton solutions with a cusp.