Multisoliton solutions of the two-component Camassa–Holm equation and its reductions

The Bäcklund transformation for an integrable two-component Camassa–Holm ($2$CH) equation is presented and studied. It involves both dependent and independent variables. A nonlinear superposition formula is given for constructing multisoliton, multiloop, and multikink solutions of the $2$CH equation. We also present solutions of the Camassa–Holm equation, the two-component Hunter–Saxton ($2$HS) equation, and the Hunter–Saxton equation, which all arise from solutions of the $2$CH equation. By appropriate limit procedures, a solution of the $2$HS equation is successfully obtained from that of the $2$CH equation, which is worked out with the method of Bäcklund transformations. By analyzing the solution, we obtain the soliton and loop solutions for $2$HS equation.