Global Existence for the Generalized Two-Component Hunter–Saxton System

Abstract

We study the global existence of solutions to a two-component generalized Hunter-Saxton system in the periodic setting. We first prove a persistence result of the solutions. Then for some particular choices of the parameters (α, κ), we show the precise blow-up scenarios and the existence of global solutions to the generalized Hunter-Saxton system under proper assumptions on the initial data. This significantly improves recent results obtained in [46,47].

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

“…The 2-HS equation has attracted much attention in the past and it has been studied extensively by many authors [9,10,11,12,13,14,15,16,17]. In a series papers by Wunsch, he studies the local and global weak solutions for the periodic 2-HS equation.…”

“…In a series papers by Wunsch, he studies the local and global weak solutions for the periodic 2-HS equation. The single solitary wave solution was studied in [13,15,17]. The local wellposedness and wave-breaking was studied by Moon and Liu [14].…”

Analytic Solutions of the Cauchy Problem for the Generalized Two-Component Hunter-Saxton System

“…Furthermore, mathematical properties of this system have been also studied further in many works, for example [10,16,17,27,28,29,30].…”

“…Lenells-Lechtenfeld [16] showed that it can be interpreted as the Euler equation on the superconformal algebra of contact vector fields on the 1|2-dimensional supercircle. Moreover, Wu-Wunsch [27], and Liu-Yin [17] gave sufficient conditions for the global existence of strong solutions to the Hunter-Saxton system. On the other hand, Escher [10] gives geometric meaning to the two-component Hunter-Saxton system, which is used by Wunsch [30] to show that there are global conservative solutions.…”

Single peak solitary wave and compacton solutions of the generalized two-component Hunter–Saxton system

“…In [10][11][12], Moon and Wu considered the generalized two-component Hunter-Saxton system with k = 1 u x xt + 2σ u x u x x + σ uu x x x − kρρ x + Au x = 0, ρ t + (ρu) x = 0, (1.2) where σ is a new free parameter and A ≥ 0. The system (1.2) is the short wave limit ((t, x) → ( t, x), → 0) of the generalized two-component Camassa-Holm system (1.1).…”