An integrable semi-discretization of the two-component Hunter-Saxton equation

Abstract

In this paper, we propose an integrable semi-discretization of the two-component Hunter-Saxton (2-HS) equation, which is obtained as the short-wave limit of the two-component Camassa-Holm (2-CH) equation. We also show that the 2-HS equation can be derived from a new set of bilinear equations, distinct from previously known ones, via a pseudo 2-reduction and a hodograph transformation. Furthermore, we construct the N-soliton solutions of both the continuous and semi-discrete systems in Wronskian and Casoratian forms, respectively.