Multi-soliton solution to the two-component Hunter–Saxton equation

• We propose a set of bilinear equations to the two-component Hunter–Saxton (2-HS) equation.
• We find one- and two-soliton solutions to the 2-HS equation by Hirota’s perturbation method.
• Through the reduction from the extended two-dimensional Toda hierarchy, we drive the multi-soliton solution to the 2-HS equation.
• The connection between the Hunter–Saxton equation and the 2-HS equation is made clear at the level of bilinear equations and multi-soliton solution.

In this paper, we study the bilinear form and the general NN-soliton solution for a two-component Hunter–Saxton (2-HS) equation, which is the short wave limit of a two-component Camassa–Holm equation. By defining a hodograph transformation based on a conservation law and appropriate dependent variable transformations, we propose a set of bilinear equations which yields the 2-HS equation. Furthermore, we construct the NN-soliton solution to the 2-HS equation based on the tau functions of an extended two-dimensional Toda-lattice hierarchy through reductions. One- and two-soliton solutions are calculated and analyzed.