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 shortwave model for the Camassa-Holm equation utxx − 2ux + 2uxuxx + uuxxx = 0 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.

Long-time asymptotics for the short pulse equation

“…In order to construct the solution u(x, t) of the CSP equation (1.1), we need use the expansion of the eigenfunction as spectral parameter k → 0. This is similar to the short pulse equation [13], the Camassa-Holm equation [15], and modified Hunter-Saxton equation [16]. Hence, in the following we use two different transformations to analyze these two singularities (k = 0 and k = ∞), respectively.…”

The Ostrovsky–Vakhnenko equation by a Riemann–Hilbert approach

“…Since the structure of the jump matrix is similar, in many aspects, to that in the case of the Degasperis-Procesi equation, the long time analysis shares many common features with that for the DP equation [8]. On the other hand, the short wave limit nature of (2.1a) suggests the presence of certain common issues with the case of the short wave Camassa-Holm equation [9].…”

“…where Θ(ζ, z) has the same form, up to sign and scaling factors, as in the case of the short wave limit of the CH equation [9]:…”

The inverse scattering transform in the form of a Riemann-Hilbert problem for the Dullin-Gottwald-Holm equation

“…Provided the RH problem is solved, the construction will give a solution of the DGH equation, that needn't be a classical solution but solves the DGH equation locally, probably in a certain week sense (see, e.g., [11] for the case of the short wave limit of the CH equation). For example, consider the RH problem with the trivial jump and with the residue conditions corresponding to N = 1 and k = iν with ν > i 2α…”