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Meromorphic integrability of the Hamiltonian systems with homogeneous potentials of degree -4
Global weak solutions to the generalized mCH equation via characteristics
| 1. | School of Mathematics, Harbin Institute of Technology, Harbin 150001, China |
| 2. | Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China |
| 3. | School of Mathematics, Harbin Institute of Technology, Harbin 150001, China |
In this paper, we study the generalized modified Camassa-Holm (gmCH) equation via characteristics. We first change the gmCH equation for unknowns $ (u,m) $ into its Lagrangian dynamics for characteristics $ X(\xi,t) $, where $ \xi\in\mathbb{R} $ is the Lagrangian label. When $ X_\xi(\xi,t)>0 $, we use the solutions to the Lagrangian dynamics to recover the classical solutions with $ m(\cdot,t)\in C_0^k(\mathbb{R}) $ ($ k\in\mathbb{N},\; \; k\geq1 $) to the gmCH equation. The classical solutions $ (u,m) $ to the gmCH equation will blow up if $ \inf_{\xi\in\mathbb{R}}X_\xi(\cdot,T_{\max}) = 0 $ for some $ T_{\max}>0 $. After the blow-up time $ T_{\max} $, we use a double mollification method to mollify the Lagrangian dynamics and construct global weak solutions (with $ m $ in space-time Radon measure space) to the gmCH equation by some space-time BV compactness arguments.
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Global $N$-peakon weak solutions to a family of nonlinear equations, J. Differential Equations, 271 (2021), 343-355.
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The modified Camassa-Holm equation in Lagarange coordinates, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 2545-2592.
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Z. Guo, X. Liu, X. Liu and C. Qu,
Stability of peakons for the generalized modified Camassa-Holm equation, J. Differential Equations, 266 (2019), 7749-7779.
doi: 10.1016/j.jde.2018.12.014. |
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X. Liu,
Orbital stability of peakons for a modified Camassa-Holm equation with higher-order nonlinearity, Discrete Contin. Dyn. Syst., 38 (2018), 5505-5521.
doi: 10.3934/dcds.2018242. |
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X. Liu, Stability in the energy space of the sum of N peakons for a modified Camassa-Holm equation with higher-order nonlinearity, J. Math. Phys., 59 (2018), 121505.
doi: 10.1063/1.5034143. |
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A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, 2002.
|
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P. J. Olver and P. Rosenau,
Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E, 53 (1996), 1900-1906.
doi: 10.1103/PhysRevE.53.1900. |
| [12] |
Z. Qiao, A new integrable equation with cuspons and W/M-shape-peaks solitons, J. Math. Phys., 47 (2006), 112701.
doi: 10.1063/1.2365758. |
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M. Yang, Y. Li and Y. Zhao,
On the Cauchy problem of generalized Fokas-Olver-Resenau-Qiao equation, Appl. Anal., 97 (2018), 2246-2268.
doi: 10.1080/00036811.2017.1359565. |
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S. Yang, Blow-up phenomena for the generalized FORQ/MCH equation, Z. Angew. Math. Phys., 71 (2020), Paper No. 20, 13 pp.
doi: 10.1007/s00033-019-1241-9. |
show all references
References:
| [1] |
S. C. Anco and E. Recio, A general family of multi-peakon equations and their properties, J. Phys. A, 52 (2019), 125203.
doi: 10.1088/1751-8121/ab03dd. |
| [2] |
A. Bressan, Hyperbolic Systems of Conservation Laws: The One-Dimensional Cauchy Problem, Oxford University Press on Demand, 2000. |
| [3] |
A. S. Fokas,
The Korteweg-de Vries equation and beyond, Acta Appl. Math., 39 (1995), 295-305.
doi: 10.1007/BF00994638. |
| [4] |
B. Fuchssteiner,
Some tricks from the symmetry-toolbox for nonlinear equations: Generalizations of the Camassa-Holm equation, Phys. D, 95 (1996), 229-243.
doi: 10.1016/0167-2789(96)00048-6. |
| [5] |
Y. Gao and H. Liu,
Global $N$-peakon weak solutions to a family of nonlinear equations, J. Differential Equations, 271 (2021), 343-355.
doi: 10.1016/j.jde.2020.08.042. |
| [6] |
Y. Gao and J.-G. Liu,
The modified Camassa-Holm equation in Lagarange coordinates, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 2545-2592.
doi: 10.3934/dcdsb.2018067. |
| [7] |
Z. Guo, X. Liu, X. Liu and C. Qu,
Stability of peakons for the generalized modified Camassa-Holm equation, J. Differential Equations, 266 (2019), 7749-7779.
doi: 10.1016/j.jde.2018.12.014. |
| [8] |
X. Liu,
Orbital stability of peakons for a modified Camassa-Holm equation with higher-order nonlinearity, Discrete Contin. Dyn. Syst., 38 (2018), 5505-5521.
doi: 10.3934/dcds.2018242. |
| [9] |
X. Liu, Stability in the energy space of the sum of N peakons for a modified Camassa-Holm equation with higher-order nonlinearity, J. Math. Phys., 59 (2018), 121505.
doi: 10.1063/1.5034143. |
| [10] |
A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, 2002.
|
| [11] |
P. J. Olver and P. Rosenau,
Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E, 53 (1996), 1900-1906.
doi: 10.1103/PhysRevE.53.1900. |
| [12] |
Z. Qiao, A new integrable equation with cuspons and W/M-shape-peaks solitons, J. Math. Phys., 47 (2006), 112701.
doi: 10.1063/1.2365758. |
| [13] |
M. Yang, Y. Li and Y. Zhao,
On the Cauchy problem of generalized Fokas-Olver-Resenau-Qiao equation, Appl. Anal., 97 (2018), 2246-2268.
doi: 10.1080/00036811.2017.1359565. |
| [14] |
S. Yang, Blow-up phenomena for the generalized FORQ/MCH equation, Z. Angew. Math. Phys., 71 (2020), Paper No. 20, 13 pp.
doi: 10.1007/s00033-019-1241-9. |
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