Multisoliton solutions of the two-component Camassa-Holm system and their reductions

We develop a systematic procedure for constructing soliton solutions of an integrable two-component Camassa-Holm (CH2) system. The parametric representation of the multisoliton solutions is obtained by using a direct method combined with a reciprocal transformation. The properties of the solutions are then investigated in detail focusing mainly on the smooth one- and two-soliton solutions. The general $N$-soliton case is described shortly. Subsequently, we show that the CH2 system reduces to the CH equation and the two-component Hunter-Saxton (HS2) system by means of appropriate limiting procedures. The corresponding expressions of the multisoliton solutions are presented in parametric forms, reproducing the existing results for the reduced equations. Last, we discuss the reduction from the HS2 system to the HS equation.