Persistence properties and unique continuation for a dispersionless two-component Camassa-Holm system with peakon and weak kink solutions

Previous Article
Limiting distribution and error terms for the number of visits to balls in non-uniformly hyperbolic dynamical systems
DCDS Home
This Issue
Next Article
Heat equation with a nonlinear boundary condition and uniformly local $L^r$ spaces

May 2016, 36(5): 2613-2625. doi: 10.3934/dcds.2016.36.2613

Persistence properties and unique continuation for a dispersionless two-component Camassa-Holm system with peakon and weak kink solutions

1.	Department of Mathematics, South China Agricultural University, 510642 Guangzhou, China
2.	Department of Mathematics, University of Texas Pan American, 78541 Edinburg, TX, United States

Received April 2015 Revised May 2015 Published October 2015

Abstract
Related Papers

In this paper, we study the persistence properties and unique continuation for a dispersionless two-component system with peakon and weak kink solutions. These properties guarantee strong solutions of the two-component system decay at infinity in the spatial variable provided that the initial data satisfies the condition of decaying at infinity. Furthermore, we give an optimal decaying index of the momentum for the system and show that the system exhibits unique continuation if the initial momentum $m_0$ and $n_0$ are non-negative.

Keywords: weak kink solutions, unique continuation., persistence properties, Two-component system, peakon.

Mathematics Subject Classification: 35G25, 35L0.

Citation: Qiaoyi Hu, Zhijun Qiao. Persistence properties and unique continuation for a dispersionless two-component Camassa-Holm system with peakon and weak kink solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2613-2625. doi: 10.3934/dcds.2016.36.2613

References:

[1]	A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation,, Arch. Ration. Mech. Anal., 183 (2007), 215. doi: 10.1007/s00205-006-0010-z. Google Scholar
[2]	R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661. doi: 10.1103/PhysRevLett.71.1661. Google Scholar
[3]	A. Constantin, The Hamiltonian structure of the Camassa-Holm equation,, Exposition. Math., 15 (1997), 53. Google Scholar
[4]	A. Constantin, On the scattering problem for the Camassa-Holm equation,, Proc. Roy. Soc. London A, 457 (2001), 953. doi: 10.1098/rspa.2000.0701. Google Scholar
[5]	A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523. doi: 10.1007/s00222-006-0002-5. Google Scholar
[6]	A. Constantin, The Cauchy problem for the periodic Camassa-Holm equation,, J. Differential Equations, 141 (1997), 218. doi: 10.1006/jdeq.1997.3333. Google Scholar
[7]	A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Ration. Mech. Anal., 192 (2009), 165. doi: 10.1007/s00205-008-0128-2. Google Scholar
[8]	A. Constantin and J. Escher, Well-posedness, global existence and blowup phenomena for a periodic quasi-linear hyperbolic equation,, Comm. Pure Appl. Math., 51 (1998), 475. doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5. Google Scholar
[9]	A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Math., 181 (1998), 229. doi: 10.1007/BF02392586. Google Scholar
[10]	A. Constantin and J. Escher, On the structure of a family of quasilinear equations arising in a shallow water theory,, Math. Ann., 312* (1998), 403. doi: 10.1007/s002080050228. Google Scholar*
[11]	A. Constantin and H. P. McKean, A shallow water equation on the circle,, Comm. Pure Appl. Math., 52 (1999), 949. doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D. Google Scholar
[12]	A. Constantin and L. Molinet, Global weak solutions for a shallow water equation,, Comm. Math. Phys., 211 (2000), 45. doi: 10.1007/s002200050801. Google Scholar
[13]	A. Constantin and W. Strauss, Stability of peakons,, Comm. Pure Appl. Math., 53 (2000), 603. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. Google Scholar
[14]	L. Escauriaza, C. E. Kenig, G. Ponce and L. Vega, On unique continuation of solutions of Schrödinger equations,, Comm. Partial Differential Equations, 31 (2006), 1811. doi: 10.1080/03605300500530446. Google Scholar
[15]	L. Escauriaza, C. E. Kenig, G. Ponce and L. Vega, On uniqueness properties of solutions of the k-generalized KdV equations,, J. Funct. Anal., 244* (2007), 504. doi: 10.1016/j.jfa.2006.11.004. Google Scholar*
[16]	A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformations and hereditary symmetries,, Phys. D, 4 (1981), 47. doi: 10.1016/0167-2789(81)90004-X. Google Scholar
[17]	A. Fokas, On a class of physically important integrable equations,, Phys. D, 87 (1995), 145. doi: 10.1016/0167-2789(95)00133-O. Google Scholar
[18]	D. Henry, Infinite propagation speed for a two component Camassa-Holm equation,, Discrete Contin. Dyn. Syst. Ser. B Appl. Algorithms, 12 (2009), 597. doi: 10.3934/dcdsb.2009.12.597. Google Scholar
[19]	A. Himonas, G. Misiolek, G. Ponce and Y. Zhou, Persistence properties and unique continuation of solutions of the Camassa-Holm equation,, Commun. Math. Phys., 271* (2007), 511. doi: 10.1007/s00220-006-0172-4. Google Scholar*
[20]	R. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves,, J. Fluid Mech., 455 (2002), 63. doi: 10.1017/S0022112001007224. Google Scholar
[21]	Y. Li and P. Oliver, Well-posedness and blow-up solutions for an integrable nonlinear dispersive model wave equation,, J. Differential Equations, 162 (2000), 27. doi: 10.1006/jdeq.1999.3683. Google Scholar
[22]	P. J. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support,, Phys. Rev. E, 53 (1996), 1900. doi: 10.1103/PhysRevE.53.1900. Google Scholar
[23]	Z. Qiao, A new integrable equation with cuspons and W/M-shape-peaks solitons,, J. Math. Phys., 47 (2006). doi: 10.1063/1.2365758. Google Scholar
[24]	Z. Qiao, New integrable hierarchy, parametric solutions, cuspons, one-peak solitons, and M/W-shape peak solutions,, J. Math. Phys., 48 (2007). doi: 10.1063/1.2759830. Google Scholar
[25]	Z. Qiao, B. Xia and J. Li, Integrable system with peakon, weak kink, and kink-peakon interactional solutions,, Front. Math. China, 8 (2013), 1185. doi: 10.1007/s11464-013-0314-x. Google Scholar
[26]	G. Rodriguez-Blanco, On the Cauchy problem for the Camassa-Holm equation,, Nonlinear Anal., 46 (2001), 309. doi: 10.1016/S0362-546X(01)00791-X. Google Scholar
[27]	X. Wu and B. Guo, Persistence properties and infinite propagation for the modified 2-component Camassa-Holm equation,, Discrete Contin. Dyn. Syst. Ser. A, 33 (2013), 3211. doi: 10.3934/dcds.2013.33.3211. Google Scholar
[28]	B. Xia and Z. Qiao, A new two-component integrable system with peakon and weak kink solutions,, Proc. R. Soc. A, 471* (2015). doi: 10.1098/rspa.2014.0750. Google Scholar*
[29]	B. Xia, Z. Qiao and R. Zhou, A synthetical integrable two-component model with peakon solutions,, , (). Google Scholar
[30]	Z. Xin and P. Zhang, On the weak solutions to a shallow water equation,, Comm. Pure Appl. Math., 53 (2000), 1411. doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5. Google Scholar
[31]	K. Yan, Z. Qiao and Z. Yin, Qualitative analysis for a new integrable two-Component Camassa-Holm system with peakon and weak kink solutions,, Commun. Math. Phys., 336* (2015), 581. doi: 10.1007/s00220-014-2236-1. Google Scholar*

show all references

References:

[1]	A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation,, Arch. Ration. Mech. Anal., 183 (2007), 215. doi: 10.1007/s00205-006-0010-z. Google Scholar
[2]	R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661. doi: 10.1103/PhysRevLett.71.1661. Google Scholar
[3]	A. Constantin, The Hamiltonian structure of the Camassa-Holm equation,, Exposition. Math., 15 (1997), 53. Google Scholar
[4]	A. Constantin, On the scattering problem for the Camassa-Holm equation,, Proc. Roy. Soc. London A, 457 (2001), 953. doi: 10.1098/rspa.2000.0701. Google Scholar
[5]	A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523. doi: 10.1007/s00222-006-0002-5. Google Scholar
[6]	A. Constantin, The Cauchy problem for the periodic Camassa-Holm equation,, J. Differential Equations, 141 (1997), 218. doi: 10.1006/jdeq.1997.3333. Google Scholar
[7]	A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Ration. Mech. Anal., 192 (2009), 165. doi: 10.1007/s00205-008-0128-2. Google Scholar
[8]	A. Constantin and J. Escher, Well-posedness, global existence and blowup phenomena for a periodic quasi-linear hyperbolic equation,, Comm. Pure Appl. Math., 51 (1998), 475. doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5. Google Scholar
[9]	A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Math., 181 (1998), 229. doi: 10.1007/BF02392586. Google Scholar
[10]	A. Constantin and J. Escher, On the structure of a family of quasilinear equations arising in a shallow water theory,, Math. Ann., 312* (1998), 403. doi: 10.1007/s002080050228. Google Scholar*
[11]	A. Constantin and H. P. McKean, A shallow water equation on the circle,, Comm. Pure Appl. Math., 52 (1999), 949. doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D. Google Scholar
[12]	A. Constantin and L. Molinet, Global weak solutions for a shallow water equation,, Comm. Math. Phys., 211 (2000), 45. doi: 10.1007/s002200050801. Google Scholar
[13]	A. Constantin and W. Strauss, Stability of peakons,, Comm. Pure Appl. Math., 53 (2000), 603. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. Google Scholar
[14]	L. Escauriaza, C. E. Kenig, G. Ponce and L. Vega, On unique continuation of solutions of Schrödinger equations,, Comm. Partial Differential Equations, 31 (2006), 1811. doi: 10.1080/03605300500530446. Google Scholar
[15]	L. Escauriaza, C. E. Kenig, G. Ponce and L. Vega, On uniqueness properties of solutions of the k-generalized KdV equations,, J. Funct. Anal., 244* (2007), 504. doi: 10.1016/j.jfa.2006.11.004. Google Scholar*
[16]	A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformations and hereditary symmetries,, Phys. D, 4 (1981), 47. doi: 10.1016/0167-2789(81)90004-X. Google Scholar
[17]	A. Fokas, On a class of physically important integrable equations,, Phys. D, 87 (1995), 145. doi: 10.1016/0167-2789(95)00133-O. Google Scholar
[18]	D. Henry, Infinite propagation speed for a two component Camassa-Holm equation,, Discrete Contin. Dyn. Syst. Ser. B Appl. Algorithms, 12 (2009), 597. doi: 10.3934/dcdsb.2009.12.597. Google Scholar
[19]	A. Himonas, G. Misiolek, G. Ponce and Y. Zhou, Persistence properties and unique continuation of solutions of the Camassa-Holm equation,, Commun. Math. Phys., 271* (2007), 511. doi: 10.1007/s00220-006-0172-4. Google Scholar*
[20]	R. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves,, J. Fluid Mech., 455 (2002), 63. doi: 10.1017/S0022112001007224. Google Scholar
[21]	Y. Li and P. Oliver, Well-posedness and blow-up solutions for an integrable nonlinear dispersive model wave equation,, J. Differential Equations, 162 (2000), 27. doi: 10.1006/jdeq.1999.3683. Google Scholar
[22]	P. J. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support,, Phys. Rev. E, 53 (1996), 1900. doi: 10.1103/PhysRevE.53.1900. Google Scholar
[23]	Z. Qiao, A new integrable equation with cuspons and W/M-shape-peaks solitons,, J. Math. Phys., 47 (2006). doi: 10.1063/1.2365758. Google Scholar
[24]	Z. Qiao, New integrable hierarchy, parametric solutions, cuspons, one-peak solitons, and M/W-shape peak solutions,, J. Math. Phys., 48 (2007). doi: 10.1063/1.2759830. Google Scholar
[25]	Z. Qiao, B. Xia and J. Li, Integrable system with peakon, weak kink, and kink-peakon interactional solutions,, Front. Math. China, 8 (2013), 1185. doi: 10.1007/s11464-013-0314-x. Google Scholar
[26]	G. Rodriguez-Blanco, On the Cauchy problem for the Camassa-Holm equation,, Nonlinear Anal., 46 (2001), 309. doi: 10.1016/S0362-546X(01)00791-X. Google Scholar
[27]	X. Wu and B. Guo, Persistence properties and infinite propagation for the modified 2-component Camassa-Holm equation,, Discrete Contin. Dyn. Syst. Ser. A, 33 (2013), 3211. doi: 10.3934/dcds.2013.33.3211. Google Scholar
[28]	B. Xia and Z. Qiao, A new two-component integrable system with peakon and weak kink solutions,, Proc. R. Soc. A, 471* (2015). doi: 10.1098/rspa.2014.0750. Google Scholar*
[29]	B. Xia, Z. Qiao and R. Zhou, A synthetical integrable two-component model with peakon solutions,, , (). Google Scholar
[30]	Z. Xin and P. Zhang, On the weak solutions to a shallow water equation,, Comm. Pure Appl. Math., 53 (2000), 1411. doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5. Google Scholar
[31]	K. Yan, Z. Qiao and Z. Yin, Qualitative analysis for a new integrable two-Component Camassa-Holm system with peakon and weak kink solutions,, Commun. Math. Phys., 336* (2015), 581. doi: 10.1007/s00220-014-2236-1. Google Scholar*

[1]	Kai Yan, Zhijun Qiao, Yufeng Zhang. On a new two-component $b$-family peakon system with cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5415-5442. doi: 10.3934/dcds.2018239
[2]	Qiaoyi Hu, Zhijun Qiao. Analyticity, Gevrey regularity and unique continuation for an integrable multi-component peakon system with an arbitrary polynomial function. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6975-7000. doi: 10.3934/dcds.2016103
[3]	Wei Luo, Zhaoyang Yin. Local well-posedness in the critical Besov space and persistence properties for a three-component Camassa-Holm system with N-peakon solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5047-5066. doi: 10.3934/dcds.2016019
[4]	Xiuting Li, Lei Zhang. The Cauchy problem and blow-up phenomena for a new integrable two-component peakon system with cubic nonlinearities. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3301-3325. doi: 10.3934/dcds.2017140
[5]	Shaojie Yang, Tianzhou Xu. Symmetry analysis, persistence properties and unique continuation for the cross-coupled Camassa-Holm system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 329-341. doi: 10.3934/dcds.2018016
[6]	Jibin Li. Bifurcations and exact travelling wave solutions of the generalized two-component Hunter-Saxton system. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1719-1729. doi: 10.3934/dcdsb.2014.19.1719
[7]	Caixia Chen, Shu Wen. Wave breaking phenomena and global solutions for a generalized periodic two-component Camassa-Holm system. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3459-3484. doi: 10.3934/dcds.2012.32.3459
[8]	Katrin Grunert. Blow-up for the two-component Camassa--Holm system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2041-2051. doi: 10.3934/dcds.2015.35.2041
[9]	Yong Chen, Hongjun Gao, Yue Liu. On the Cauchy problem for the two-component Dullin-Gottwald-Holm system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3407-3441. doi: 10.3934/dcds.2013.33.3407
[10]	José G. Llorente. Mean value properties and unique continuation. Communications on Pure & Applied Analysis, 2015, 14 (1) : 185-199. doi: 10.3934/cpaa.2015.14.185
[11]	Piotr Bogusław Mucha, Milan Pokorný, Ewelina Zatorska. Approximate solutions to a model of two-component reactive flow. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 1079-1099. doi: 10.3934/dcdss.2014.7.1079
[12]	Yongsheng Mi, Boling Guo, Chunlai Mu. Well-posedness and blow-up scenario for a new integrable four-component system with peakon solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2171-2191. doi: 10.3934/dcds.2016.36.2171
[13]	Jingqun Wang, Lixin Tian, Weiwei Guo. Global exact controllability and asympotic stabilization of the periodic two-component $\mu\rho$-Hunter-Saxton system. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2129-2148. doi: 10.3934/dcdss.2016088
[14]	Huijun He, Zhaoyang Yin. On the Cauchy problem for a generalized two-component shallow water wave system with fractional higher-order inertia operators. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1509-1537. doi: 10.3934/dcds.2017062
[15]	Kai Yan, Zhaoyang Yin. Well-posedness for a modified two-component Camassa-Holm system in critical spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1699-1712. doi: 10.3934/dcds.2013.33.1699
[16]	Zeng Zhang, Zhaoyang Yin. Global existence for a two-component Camassa-Holm system with an arbitrary smooth function. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5523-5536. doi: 10.3934/dcds.2018243
[17]	Vural Bayrak, Emil Novruzov, Ibrahim Ozkol. Local-in-space blow-up criteria for two-component nonlinear dispersive wave system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 6023-6037. doi: 10.3934/dcds.2019263
[18]	Mouhamed Moustapha Fall, Veronica Felli. Unique continuation properties for relativistic Schrödinger operators with a singular potential. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5827-5867. doi: 10.3934/dcds.2015.35.5827
[19]	Christian Klingenberg, Marlies Pirner, Gabriella Puppo. A consistent kinetic model for a two-component mixture with an application to plasma. Kinetic & Related Models, 2017, 10 (2) : 445-465. doi: 10.3934/krm.2017017
[20]	Qiaoyi Hu, Zhixin Wu, Yumei Sun. Liouville theorems for periodic two-component shallow water systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 3085-3097. doi: 10.3934/dcds.2018134

2018 Impact Factor: 1.143

American Institute of Mathematical Sciences