A two-component μ-Hunter–Saxton equation

Zuo, Dafeng

doi:10.1088/0266-5611/26/8/085003

Cited by 38 publications

(39 citation statements)

References 26 publications

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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%

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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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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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“…It is well known that the KdV equation could be regarded as Euler equations on the dual of Virasoro algebra vir with different inner products ( [7,9,11,12]). Let us remark that V.I.Arnold in [1] suggested a general framework for the Euler equation on an arbitrary Lie group G, which is useful to characterize a variety of conservative dynamical systems, please see e.g., [2,6,7,8,9,11,12,15,17,18] and references therein. If the corresponding Lie algebra is G, then the Euler equation (1.1) on G * could describe a geodesic flow w.r.t a suitable one-side invariant Riemannian metric on Lie group G.…”

Section: Introductionmentioning

confidence: 99%

The Frobenius–Virasoro algebra and Euler equations

Zuo

2014

Journal of Geometry and Physics

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“…In this paper, we consider the Cauchy problem of the following two‐component periodic μ ‐Hunter–Saxton system

({, \begin{matrix} μ (u) (t) - u_{txx} = 2 μ (u) u_{x} - 2 u_{x} u_{xx} - u u_{xxx} + ρ ρ_{x} - γ_{1} u_{xxx}, & t > 0, x \in R, \\ ρ_{t} = {(ρu)}_{x} - 2 γ_{2} ρ_{x}, & t > 0, x \in R, \\ u (0, x) = u_{0} (x), ρ (0, x) = ρ_{0} (x), & x \in R, \\ u (t, x + 1) = u (t, x), ρ (t, x + 1) = ρ (t, x), & t \geq 0, x \in R, \end{matrix})

where

μ (u) = \int_{double-struckS} u normald x

double-struckS = double-struckR / double-struckZ

and

γ_{i} \in double-struckR

, i = 1,2. The system was introduced by Zou . By integrating both sides of the first equation in the system over the circle

double-struckS

and using the periodicity of u , one obtains

μ (u_{t}) = μ {(u)}_{t} = 0,

…”

Section: Introductionmentioning

confidence: 99%

“…By integrating both sides of the first equation in the system over the circle

double-struckS

and using the periodicity of u , one obtains

μ (u_{t}) = μ {(u)}_{t} = 0,

which implies the following two‐component periodic μ ‐Hunter–Saxton system

({, \begin{matrix} - u_{txx} = 2 μ (u) u_{x} - 2 u_{x} u_{xx} - u u_{xxx} + ρ ρ_{x} - γ_{1} u_{xxx}, & t > 0, x \in R, \\ ρ_{t} = {(ρu)}_{x} - 2 γ_{2} ρ_{x}, & t > 0, x \in R, \\ u (0, x) = u_{0} (x), ρ (0, x) = ρ_{0} (x), & x \in R, \\ u (t, x + 1) = u (t, x), ρ (t, x + 1) = ρ (t, x), & t \geq 0, x \in R . \end{matrix})

This system is a two‐component generalization of the generalized Hunter–Saxton equation obtained in . Zou shows that this system is both a bi‐Hamiltonian Euler equation and a bi‐variational equation. Liu and Yin established the local well‐posedness, precise blow‐up scenario and global existence result to the system .…”

Section: Introductionmentioning

confidence: 99%