The curvature of semidirect product groups associated with two-component Hunter–Saxton systems

Kohlmann, Martin

doi:10.1088/1751-8113/44/22/225203

Cited by 13 publications

(20 citation statements)

References 40 publications

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“…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.…”

Section: Introductionmentioning

confidence: 99%

“…The local wellposedness and wave-breaking was studied by Moon and Liu [14]. The geometric property of the 2-HS equation was investigated in [16]. The two-component Hunter Saxton system is a particular case of the Gurevich Zybin system pertaining to nonlinear one-dimensional dynamics of dark matter as well as nonlinear ion-acoustic waves (cf.…”

Section: Introductionmentioning

confidence: 99%

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

2016

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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.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

2016

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“…It was shown that the system describes the geodesic flows on semidirect product Lie group of diffeomorphism group D s .S/ and the space of scalar functions on S [17,18]. One can readily verify that it also describes pseudo-spherical surfaces [19], namely there exist forms associated with the system…”

Section: Y Zhangmentioning

confidence: 94%

“…System can be viewed as a mid‐way one between the two‐component Camassa–Holm system and two‐component Hunter–Saxton system, which admits the Lax‐pair

\begin{matrix} aligned \\ right φ_{xx} & left = (λm + λ^{2} ρ^{2}) φ, & right \\ right φ_{t} & left = (\frac{1}{2 λ} - u) φ_{x} + \frac{1}{2} u_{x} φ . \end{matrix}

It was shown that the system describes the geodesic flows on semidirect product Lie group of diffeomorphism group

{scriptD}^{s} (double-struckS)

and the space of scalar functions on

double-struckS

. One can readily verify that it also describes pseudo‐spherical surfaces , namely there exist forms associated with the system

\begin{matrix} aligned \\ right ω_{1} & left = (\frac{1}{4} λ^{2} MathClass-open(ρ^{2} - 1 MathClass-close) + \frac{1}{2} λm + 1) dx + (\frac{1}{4} λ^{2} u MathClass-open(1 - ρ^{2} MathClass-close) - \frac{1}{4} λ (1 + 2 um - ρ^{2} + 2 u_{x}) + \frac{1}{2} μ - u) dt, & right \\ right ω_{2} & left = \end{matrix}

…”

Section: Introductionmentioning

confidence: 99%

Global weak solutions for a periodic two‐component μ‐Camassa–Holm system

Zhang

2013

Math Methods in App Sciences

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In this paper, we consider the global existence of weak solutions for a two‐component μ‐Camassa–Holm system in the periodic setting. Global existence for strong solutions to the system with smooth approximate initial value is derived. Then, we show that the limit of approximate solutions is a global‐in‐time weak solution of the two‐component μ‐Camassa–Holm system. Copyright © 2013 John Wiley & Sons, Ltd.

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“…When ρ ≢0 and γ i =0( i = 1,2), Constantin and Ivanov considered the peakon solutions of the Cauchy problem of system . In paper , Wunsch studied the Cauchy problem of two‐component periodic Hunter–Saxton system; see also .…”

Section: Introductionmentioning

confidence: 99%