Qualitative analysis for a new integrable two-component Camassa-Holm system with peakon and weak kink solutions
type: Item
instance of: scholarly article
title: Qualitative analysis for a new integrable two-component Camassa-Holm system with peakon and weak kink solutions
zbMATH Open document ID: 1317.35219
author: Kai Yan
author: Zhijun Qiao
author: Zhaoyang Yin
published in: Communications in Mathematical Physics
publication date: 2015-04-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/1306.0417
review text: In this paper, authors consider the following integrable two-component Camassa-Holm system with both quadratic and cubic nonlinearity problem: \[ \begin{cases} &m_t+\frac12[(uv-u_xv_x)m]_x-\frac12[(uv_x-u_xv)m]+bu_x=0,\\ &n_t+\frac12[(uv-u_xv_x)n]_x+\frac12[(uv_x-u_xv)n]+bv_x=0, \end{cases} \] where \(m = u -u_{xx}\), \(n = v -v_{xx}\) , and \(b\) takes an arbitrary value. It is the first integrable system that admits weak kink and kink-peakon interactional solutions. It can be reduced to the CH equation, the cubic CH equation. They first establish the local well-posedness for the Cauchy problem of the system, and then derive a precise blow-up scenario and a new blow-up result for strong solutions to the system with both quadratic and cubic nonlinearity. Furthermore, its peakon and weak kink solutions are discussed as well.
Mathematics Subject Classification ID: 35Q51
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 35C08
Mathematics Subject Classification ID: 35B44
Mathematics Subject Classification ID: 35D35
zbMATH DE Number: 6424613
zbMATH Keywords: two-component Camassa-Holm system
zbMATH Keywords: peakon
zbMATH Keywords: weak kink solutions
reviewed by: Qin Meng Zhao
MaRDI profile type: Publication
OpenAlex ID: W2168607175
arXiv ID: 1306.0417
cites work: Fourier Analysis and Nonlinear Partial Differential Equations
cites work: Global conservative solutions of the Camassa-Holm equation
cites work: An integrable shallow water equation with peaked solitons
cites work: Q4311808
cites work: A two-component generalization of the Camassa-Holm equation and its solutions
cites work: On the well-posedness of the Degasperis-Procesi equation
cites work: Existence of permanent and breaking waves for a shallow water equation: a geometric approach
cites work: Q4225587
cites work: Q4253188
cites work: Wave breaking for nonlinear nonlocal shallow water equations
cites work: Analyticity of periodic traveling free surface water waves with vorticity
cites work: On an integrable two-component Camassa-Holm shallow water system
cites work: Inverse scattering transform for the Degasperis–Procesi equation
cites work: Geodesic flow on the diffeomorphism group of the circle
cites work: The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations
cites work: Global weak solutions for a shallow water equation
cites work: Stability of peakons
cites work: Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod
cites work: A few remarks on the Camassa-Holm equation.
cites work: Closed-loop persistent identification of linear systems with unmodeled dynamics and stochastic disturbances
cites work: Q4508328
cites work: The Degasperis-Procesi equation as a non-metric Euler equation
cites work: Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation
cites work: Global weak solutions and blow-up structure for the Degasperis-Procesi equation
cites work: Shock waves and blow-up phenomena for the periodic Degasperis-Procesi equation
cites work: Initial boundary value problems for nonlinear dispersive wave equations
cites work: On a class of physically important integrable equations
cites work: Symplectic structures, their Bäcklund transformations and hereditary symmetries
cites work: Some tricks from the symmetry-toolbox for nonlinear equations: Generalizations of the Camassa-Holm equation
cites work: Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system
cites work: Q3079639
cites work: Global weak solutions for a two-component Camassa-Holm shallow water system
cites work: Global weak solutions for a modified two-component Camassa-Holm equation
cites work: On the global existence and wave-breaking criteria for the two-component Camassa-Holm system
cites work: Wave-breaking and peakons for a modified Camassa-Holm equation
cites work: The Cauchy problem for the Novikov equation
cites work: Integrable peakon equations with cubic nonlinearity
cites work: Algebro-geometric solutions for the Degasperis-Procesi hierarchy
cites work: Camassa–Holm, Korteweg–de Vries and related models for water waves
cites work: Q4075341
cites work: Global existence and blow-up phenomena for the Degasperis-Procesi equation
cites work: Formation and dynamics of shock waves in the Degasperis-Procesi equation
cites work: Generalizations of the Camassa–Holm equation
cites work: A two-component generalization of the Degasperis–Procesi equation
cites work: The Camassa-Holm hierarchy, \(N\)-dimensional integrable systems, and algebro-geometric solution on a symplectic submanifold
cites work: A new integrable equation with cuspons and W/M-shape-peaks solitons
cites work: Integrable peakon systems with weak kink and kink-peakon interactional solutions
cites work: Integrable hierarchy, \(3\times 3\) constrained systems, and parametric solutions
cites work: Global periodic conservative solutions of a periodic modified two-component Camassa-Holm equation
cites work: Stokes waves
cites work: Q4151274
cites work: Global weak solutions for the Novikov equation
cites work: A new two-component integrable system with peakon solutions
cites work: On the weak solutions to a shallow water equation.
cites work: Analytic solutions of the Cauchy problem for two-component shallow water systems
cites work: On the Cauchy problem for a two-component Degasperis-Procesi system

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cites work: Initial boundary value problems for the two-component shallow water systems

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cites work: On the Cauchy problem for an integrable equation with peakon solutions

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Qualitative analysis for a new integrable two-component Camassa-Holm system with peakon and weak kink solutions (DOI)
DOI: 10.1007/S00220-014-2236-1