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

TL;DR

This paper derives the N-soliton solutions for a two-component Hunter-Saxton equation using bilinear forms and tau functions, advancing understanding of its integrable structure and soliton interactions.

Contribution

It introduces a bilinear form and constructs explicit N-soliton solutions for the 2-HS equation via reductions from the Toda-lattice hierarchy.

Findings

Explicit one- and two-soliton solutions are obtained.

The N-soliton solutions are expressed through tau functions.

The method links the 2-HS equation to integrable hierarchies.

Abstract

In this paper, we study the bilinear form and the general N-soliton solution for a two-component Hunter-Saxton (2-HS) equation, which is the short wave limit of a twocomponent 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 N-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.