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

Abstract— 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.