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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

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.
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 and Continuous Dynamical Systems, 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), 215-239. doi: 10.1007/s00205-006-0010-z. [2] R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661. [3] A. Constantin, The Hamiltonian structure of the Camassa-Holm equation, Exposition. Math., 15 (1997), 53-85. [4] A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. Roy. Soc. London A, 457 (2001), 953-970. doi: 10.1098/rspa.2000.0701. [5] A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5. [6] A. Constantin, The Cauchy problem for the periodic Camassa-Holm equation, J. Differential Equations, 141 (1997), 218-235. doi: 10.1006/jdeq.1997.3333. [7] A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186. doi: 10.1007/s00205-008-0128-2. [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), 475-504. doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5. [9] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586. [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), 403-416. doi: 10.1007/s002080050228. [11] A. Constantin and H. P. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1999), 949-982. doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D. [12] A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61. doi: 10.1007/s002200050801. [13] A. Constantin and W. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. [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), 1811-1823. doi: 10.1080/03605300500530446. [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), 504-535. doi: 10.1016/j.jfa.2006.11.004. [16] A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D, 4 (1981), 47-66. doi: 10.1016/0167-2789(81)90004-X. [17] A. Fokas, On a class of physically important integrable equations, Phys. D, 87 (1995), 145-150. doi: 10.1016/0167-2789(95)00133-O. [18] D. Henry, Infinite propagation speed for a two component Camassa-Holm equation, Discrete Contin. Dyn. Syst. Ser. B Appl. Algorithms, 12 (2009), 597-606. doi: 10.3934/dcdsb.2009.12.597. [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), 511-522. doi: 10.1007/s00220-006-0172-4. [20] R. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82. doi: 10.1017/S0022112001007224. [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), 27-63. doi: 10.1006/jdeq.1999.3683. [22] P. J. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E, 53 (1996), 1900-1906. doi: 10.1103/PhysRevE.53.1900. [23] Z. Qiao, A new integrable equation with cuspons and W/M-shape-peaks solitons, J. Math. Phys., 47 (2006), 112701, 9pp. doi: 10.1063/1.2365758. [24] Z. Qiao, New integrable hierarchy, parametric solutions, cuspons, one-peak solitons, and M/W-shape peak solutions, J. Math. Phys., 48 (2007), 082701, 20pp. doi: 10.1063/1.2759830. [25] Z. Qiao, B. Xia and J. Li, Integrable system with peakon, weak kink, and kink-peakon interactional solutions, Front. Math. China, 8 (2013), 1185-1196. doi: 10.1007/s11464-013-0314-x. [26] G. Rodriguez-Blanco, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal., 46 (2001), 309-327. doi: 10.1016/S0362-546X(01)00791-X. [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), 3211-3223. doi: 10.3934/dcds.2013.33.3211. [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. [29] B. Xia, Z. Qiao and R. Zhou, A synthetical integrable two-component model with peakon solutions, arXiv:1301.3216. [30] Z. Xin and P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53 (2000), 1411-1433. doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5. [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), 581-617. doi: 10.1007/s00220-014-2236-1.

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References:
 [1] A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239. doi: 10.1007/s00205-006-0010-z. [2] R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661. [3] A. Constantin, The Hamiltonian structure of the Camassa-Holm equation, Exposition. Math., 15 (1997), 53-85. [4] A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. Roy. Soc. London A, 457 (2001), 953-970. doi: 10.1098/rspa.2000.0701. [5] A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5. [6] A. Constantin, The Cauchy problem for the periodic Camassa-Holm equation, J. Differential Equations, 141 (1997), 218-235. doi: 10.1006/jdeq.1997.3333. [7] A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186. doi: 10.1007/s00205-008-0128-2. [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), 475-504. doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5. [9] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586. [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), 403-416. doi: 10.1007/s002080050228. [11] A. Constantin and H. P. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1999), 949-982. doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D. [12] A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61. doi: 10.1007/s002200050801. [13] A. Constantin and W. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. [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), 1811-1823. doi: 10.1080/03605300500530446. [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), 504-535. doi: 10.1016/j.jfa.2006.11.004. [16] A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D, 4 (1981), 47-66. doi: 10.1016/0167-2789(81)90004-X. [17] A. Fokas, On a class of physically important integrable equations, Phys. D, 87 (1995), 145-150. doi: 10.1016/0167-2789(95)00133-O. [18] D. Henry, Infinite propagation speed for a two component Camassa-Holm equation, Discrete Contin. Dyn. Syst. Ser. B Appl. Algorithms, 12 (2009), 597-606. doi: 10.3934/dcdsb.2009.12.597. [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), 511-522. doi: 10.1007/s00220-006-0172-4. [20] R. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82. doi: 10.1017/S0022112001007224. [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), 27-63. doi: 10.1006/jdeq.1999.3683. [22] P. J. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E, 53 (1996), 1900-1906. doi: 10.1103/PhysRevE.53.1900. [23] Z. Qiao, A new integrable equation with cuspons and W/M-shape-peaks solitons, J. Math. Phys., 47 (2006), 112701, 9pp. doi: 10.1063/1.2365758. [24] Z. Qiao, New integrable hierarchy, parametric solutions, cuspons, one-peak solitons, and M/W-shape peak solutions, J. Math. Phys., 48 (2007), 082701, 20pp. doi: 10.1063/1.2759830. [25] Z. Qiao, B. Xia and J. Li, Integrable system with peakon, weak kink, and kink-peakon interactional solutions, Front. Math. China, 8 (2013), 1185-1196. doi: 10.1007/s11464-013-0314-x. [26] G. Rodriguez-Blanco, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal., 46 (2001), 309-327. doi: 10.1016/S0362-546X(01)00791-X. [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), 3211-3223. doi: 10.3934/dcds.2013.33.3211. [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. [29] B. Xia, Z. Qiao and R. Zhou, A synthetical integrable two-component model with peakon solutions, arXiv:1301.3216. [30] Z. Xin and P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53 (2000), 1411-1433. doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5. [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), 581-617. doi: 10.1007/s00220-014-2236-1.
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