# American Institute of Mathematical Sciences

December  2016, 36(12): 6975-7000. doi: 10.3934/dcds.2016103

## Analyticity, Gevrey regularity and unique continuation for an integrable multi-component peakon system with an arbitrary polynomial function

 1 Department of Mathematics, South China Agricultural University, 510642 Guangzhou 2 School of Mathematical and Statistical Sciences, University of Texas - Rio Grande Valley, 78539 Edinburg, TX, United States

Received  January 2016 Revised  March 2016 Published  October 2016

In this paper, we study the Cauchy problem for an integrable multi-component ($2N$-component) peakon system which is involved in an arbitrary polynomial function. Based on a generalized Ovsyannikov type theorem, we first prove the existence and uniqueness of solutions for the system in the Gevrey-Sobolev spaces with the lower bound of the lifespan. Then we show the continuity of the data-to-solution map for the system. Furthermore, by introducing a family of continuous diffeomorphisms of a line and utilizing the fine structure of the system, we demonstrate the system exhibits unique continuation.
Citation: 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
##### References:
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Google Scholar [6] 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 [7] A. Constantin, The Hamiltonian structure of the Camassa-Holm equation,, Exposition. Math., 15 (1997), 53. Google Scholar [8] 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 [9] A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523. doi: 10.1007/s00222-006-0002-5. Google Scholar [10] 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 [11] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Math., 181 (1998), 229. doi: 10.1007/BF02392586. Google Scholar [12] 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 [13] A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity,, Ann. of Math., 173 (2011), 559. Google Scholar [14] 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 [15] A. Constantin and J. Escher, Global existence of solutions and blow-up for a shallow water equation: A geometric approach,, Ann. Sc. Norm. Super. Pisa Cl. Sci., 26 (1998), 303. Google Scholar [16] A. Constantin and J. Escher, Global existence of solutions and breaking waves for a shallow water equation: A geometric approach,, Annales de l'Institut Fouriter (Grenoble), 50 (2000), 321. doi: 10.5802/aif.1757. Google Scholar [17] A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity,, Ann. of Math., 173 (2011), 559. doi: 10.4007/annals.2011.173.1.12. Google Scholar [18] 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 [19] 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 [20] 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 [21] A. Constantin and R. Ivanov, On an integrable two-component Camassa-Holm shallow water system,, Phys. Lett. A, 372 (2008), 7129. doi: 10.1016/j.physleta.2008.10.050. Google Scholar [22] C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations,, J. Funct. Anal., 87 (1989), 359. doi: 10.1016/0022-1236(89)90015-3. Google Scholar [23] 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 [24] A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformations and hereditary symmetries,, Phys. D, 4 (): 47. doi: 10.1016/0167-2789(81)90004-X. Google Scholar [25] F. Gesztesy and H. Holden, Soliton Equations and Their Algebro-Geometric Solutions,, Cambridge Studies in Advanced Mathematics, (2003). doi: 10.1017/CBO9780511546723. Google Scholar [26] X. Geng and B. Xue, A three-component generalization of Camassa-Holm equation with N-peakon solutions,, Adv. Math., 226 (2011), 827. doi: 10.1016/j.aim.2010.07.009. Google Scholar [27] A. Himonas and G. Misiolek, Analyticity of the Cauchy problem for an integrable evolution equation,, Math. Ann., 327 (2003), 575. doi: 10.1007/s00208-003-0466-1. Google Scholar [28] Q. Hu and Z. Qiao, Persistence properties and unique continuation for a dispersionless two-component Camassa-Holm system with peakon and weak kink solutions,, Discrete Contin. Dyn. Syst. Ser. A, 36 (2016), 2613. doi: 10.3934/dcds.2016.36.2613. Google Scholar [29] D. Holm, L. Náraigh and C. Tronci, Singular solutions of a modified two-component Camassa-Holm equation,, Phys. Rev. E, 79 (2009). doi: 10.1103/PhysRevE.79.016601. Google Scholar [30] 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 [31] 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 [32] N. Li, Q. Liu and Z. Popowicz, A four-component Camassa-Holm type hierarchy,, J. Geom. Phys., 85 (2014), 29. doi: 10.1016/j.geomphys.2014.05.026. Google Scholar [33] W. Luo and Z. Yin, Gevrey regularity and analyticity for Camassa-Holm type systems,, , (2015). Google Scholar [34] L. Nirenberg, An abstract form of the nonlinear Cauchy-Kowalevski theorem,, J. Differential Geom., 6 (1972), 561. Google Scholar [35] T. Nishida, A note on a theorem of Nirenberg,, J. Differential Geom., 12 (1977), 629. doi: projecteuclid.org/euclid.jdg/1214434231. Google Scholar [36] L. Ovsyannikov, Non-local Cauchy problems in fluid dynamics,, Actes Congress Int. Math. Nice, 3 (1971), 137. Google Scholar [37] L. Ovsyannikov, A nonlinear Cauchy problems in a scale of Banach spaces,, Dokl. Akad. Nauk. SSSR, 200 (1971), 789. Google Scholar [38] 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 [39] Z. Qiao, The Camassa-Holm hierarchy, related $N$-dimensional integrable systems and algebro-geometric solution on a symplectic submanifold,, Commun. Math. Phys., 239 (2003), 309. doi: 10.1007/s00220-003-0880-y. Google Scholar [40] 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 [41] 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 [42] Z. Qiao and B. Xia, 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 [43] 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 [44] J. Song, C. Qu and Z. Qiao, A new integrable two-component system with cubic nonlinearity,, J. Math. Phys., 52 (2011). doi: 10.1063/1.3530865. Google Scholar [45] F. Treves, Ovsyannikov theorem and hyperdifferential operators,, Conselho Nacional de Pesquisas, (1968). Google Scholar [46] F. Treves, An abstract nonlinear Cauchy-Kovalevska theorem,, Trans. Amer. Math. Soc., 150 (1970), 77. doi: 10.1090/S0002-9947-1970-0274911-X. Google Scholar [47] F. Treves, TransOvcyannikov Analyticity and Applications,, talk at VI Geometric Analysis of PDEs and Several Complex Variables, (2011). Google Scholar [48] 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 [49] B. Xia and Z. Qiao, Multi-component generalization of the Camassa-Holm equation,, J. Geom. Phys., 107 (2016), 35. doi: 10.1016/j.geomphys.2016.04.020. Google Scholar [50] B. Xia, Z. Qiao and R. Zhou, A synthetical integrable two-component model with peakon solutions,, Stud. Appl. Math., 135 (2015), 248. doi: 10.1111/sapm.12085. Google Scholar [51] B. Xia, Z. Qiao 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 [52] 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 [53] 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 [54] K. Yan and Z. Yin, Analytic solutions of the Cauchy problem for two-component shallow water systems,, Math. Z., 269 (2011), 1113. doi: 10.1007/s00209-010-0775-5. Google Scholar [55] K. Yan, Z. Qiao and Y. Zhang, Blow-up phenomena for an integrable two-component Camassa-Holm system with cubic nonlinearity and peakon solutions,, J. Differential Equations, 259 (2015), 6644. doi: 10.1016/j.jde.2015.08.004. Google Scholar [56] Z. Zhang and Z. Yin, On the Cauchy problem for a four-component Camassa-Holm type system,, Discrete Contin. Dyn. Syst. Ser. A, 35 (2015), 5153. doi: 10.3934/dcds.2015.35.5153. Google Scholar [57] Z. Zhang and Z. Yin, Well-posedness, global existence and blow-up phenomena for an integrable multi-component Camassa-Holm system,, Nonlinear Analysis, 142 (2016), 112. doi: 10.1016/j.na.2016.04.004. Google Scholar

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##### References:
 [1] S. Baouendi and C. Goulaouic, Remarks on the abstract form of nonlinear Cauchy-Kowalevski theorems,, Comm. Partial Differential Equations, 2 (1977), 1151. doi: 10.1080/03605307708820057. Google Scholar [2] S. Baouendi and C. Goulaouic, Sharp estimates for analytic pseudodifferential operators and applications to Cauchy problems,, J. Differential Equations, 48 (1983), 241. doi: 10.1016/0022-0396(83)90051-7. Google Scholar [3] R. Barostichi, A. Himonas and G. Petronilho, A Cauchy-Kovalevsky theorem for a nonlinear and nonlocal equations,, Analysis and Geometry, (2015), 59. doi: 10.1007/978-3-319-17443-3_5. Google Scholar [4] R. Barostichi, A. Himonas and G. Petronilho, Autonomous Ovsyannikov theorem and applications to nonlocal evolution equations and systems,, J. Funct. Anal., 270 (2016), 330. Google Scholar [5] 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 [6] 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 [7] A. Constantin, The Hamiltonian structure of the Camassa-Holm equation,, Exposition. Math., 15 (1997), 53. Google Scholar [8] 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 [9] A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523. doi: 10.1007/s00222-006-0002-5. Google Scholar [10] 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 [11] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Math., 181 (1998), 229. doi: 10.1007/BF02392586. Google Scholar [12] 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 [13] A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity,, Ann. of Math., 173 (2011), 559. Google Scholar [14] 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 [15] A. Constantin and J. Escher, Global existence of solutions and blow-up for a shallow water equation: A geometric approach,, Ann. Sc. Norm. Super. Pisa Cl. Sci., 26 (1998), 303. Google Scholar [16] A. Constantin and J. Escher, Global existence of solutions and breaking waves for a shallow water equation: A geometric approach,, Annales de l'Institut Fouriter (Grenoble), 50 (2000), 321. doi: 10.5802/aif.1757. Google Scholar [17] A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity,, Ann. of Math., 173 (2011), 559. doi: 10.4007/annals.2011.173.1.12. Google Scholar [18] 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 [19] 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 [20] 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 [21] A. Constantin and R. Ivanov, On an integrable two-component Camassa-Holm shallow water system,, Phys. Lett. A, 372 (2008), 7129. doi: 10.1016/j.physleta.2008.10.050. Google Scholar [22] C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations,, J. Funct. Anal., 87 (1989), 359. doi: 10.1016/0022-1236(89)90015-3. Google Scholar [23] 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 [24] A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformations and hereditary symmetries,, Phys. D, 4 (): 47. doi: 10.1016/0167-2789(81)90004-X. Google Scholar [25] F. Gesztesy and H. Holden, Soliton Equations and Their Algebro-Geometric Solutions,, Cambridge Studies in Advanced Mathematics, (2003). doi: 10.1017/CBO9780511546723. Google Scholar [26] X. Geng and B. Xue, A three-component generalization of Camassa-Holm equation with N-peakon solutions,, Adv. Math., 226 (2011), 827. doi: 10.1016/j.aim.2010.07.009. Google Scholar [27] A. Himonas and G. Misiolek, Analyticity of the Cauchy problem for an integrable evolution equation,, Math. Ann., 327 (2003), 575. doi: 10.1007/s00208-003-0466-1. Google Scholar [28] Q. Hu and Z. Qiao, Persistence properties and unique continuation for a dispersionless two-component Camassa-Holm system with peakon and weak kink solutions,, Discrete Contin. Dyn. Syst. Ser. A, 36 (2016), 2613. doi: 10.3934/dcds.2016.36.2613. Google Scholar [29] D. Holm, L. Náraigh and C. Tronci, Singular solutions of a modified two-component Camassa-Holm equation,, Phys. Rev. E, 79 (2009). doi: 10.1103/PhysRevE.79.016601. Google Scholar [30] 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 [31] 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 [32] N. Li, Q. Liu and Z. Popowicz, A four-component Camassa-Holm type hierarchy,, J. Geom. Phys., 85 (2014), 29. doi: 10.1016/j.geomphys.2014.05.026. Google Scholar [33] W. Luo and Z. Yin, Gevrey regularity and analyticity for Camassa-Holm type systems,, , (2015). Google Scholar [34] L. Nirenberg, An abstract form of the nonlinear Cauchy-Kowalevski theorem,, J. Differential Geom., 6 (1972), 561. Google Scholar [35] T. Nishida, A note on a theorem of Nirenberg,, J. Differential Geom., 12 (1977), 629. doi: projecteuclid.org/euclid.jdg/1214434231. Google Scholar [36] L. Ovsyannikov, Non-local Cauchy problems in fluid dynamics,, Actes Congress Int. Math. Nice, 3 (1971), 137. Google Scholar [37] L. Ovsyannikov, A nonlinear Cauchy problems in a scale of Banach spaces,, Dokl. Akad. Nauk. SSSR, 200 (1971), 789. Google Scholar [38] 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 [39] Z. Qiao, The Camassa-Holm hierarchy, related $N$-dimensional integrable systems and algebro-geometric solution on a symplectic submanifold,, Commun. Math. Phys., 239 (2003), 309. doi: 10.1007/s00220-003-0880-y. Google Scholar [40] 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 [41] 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 [42] Z. Qiao and B. Xia, 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 [43] 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 [44] J. Song, C. Qu and Z. Qiao, A new integrable two-component system with cubic nonlinearity,, J. Math. Phys., 52 (2011). doi: 10.1063/1.3530865. Google Scholar [45] F. Treves, Ovsyannikov theorem and hyperdifferential operators,, Conselho Nacional de Pesquisas, (1968). Google Scholar [46] F. Treves, An abstract nonlinear Cauchy-Kovalevska theorem,, Trans. Amer. Math. Soc., 150 (1970), 77. doi: 10.1090/S0002-9947-1970-0274911-X. Google Scholar [47] F. Treves, TransOvcyannikov Analyticity and Applications,, talk at VI Geometric Analysis of PDEs and Several Complex Variables, (2011). Google Scholar [48] 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 [49] B. Xia and Z. 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