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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 & 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
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