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, 2016, 36 (12) : 6975-7000. doi: 10.3934/dcds.2016103
References:
[1]

S. Baouendi and C. Goulaouic, Remarks on the abstract form of nonlinear Cauchy-Kowalevski theorems, Comm. Partial Differential Equations, 2 (1977), 1151-1162. 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), 241-268. 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, 127 of the series Springer Proceedings in Mathematics & Statistics, (2015), 59-68. 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), 330-358. Google Scholar

[5]

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.  Google Scholar

[6]

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.  Google Scholar

[7]

A. Constantin, The Hamiltonian structure of the Camassa-Holm equation, Exposition. Math., 15 (1997), 53-85.  Google Scholar

[8]

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.  Google Scholar

[9]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. 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), 218-235. 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), 229-243. 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), 475-504. 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), 559-568.  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), 403-416. 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), 303-328.  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), 321-362. 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), 559-568. 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), 165-186. 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), 949-982. 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), 45-61. 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), 7129-7132. 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), 359-369. 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), 145-150. 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, Volume79, Cambridge University Press, 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), 827-839. 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), 575-584. 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), 2613-2625. 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), 016601, 13pp. 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), 63-82. 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), 27-63. 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), 29-39. 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, http://arxiv.org/abs/1507.05250, Jul. 2015. Google Scholar

[34]

L. Nirenberg, An abstract form of the nonlinear Cauchy-Kowalevski theorem, J. Differential Geom., 6 (1972), 561-576,  Google Scholar

[35]

T. Nishida, A note on a theorem of Nirenberg, J. Differential Geom., 12 (1977), 629-633. 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), 137-142.  Google Scholar

[37]

L. Ovsyannikov, A nonlinear Cauchy problems in a scale of Banach spaces, Dokl. Akad. Nauk. SSSR, 200 (1971), 789-792.  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), 1900-1906. 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), 309-341. 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), 112701, 9pp. 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), 082701, 20pp. 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), 1185-1196. 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), 309-327. 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) 013503 (9 pages). doi: 10.1063/1.3530865.  Google Scholar

[45]

F. Treves, Ovsyannikov theorem and hyperdifferential operators, Conselho Nacional de Pesquisas, Rio de Janeiro, 1968.  Google Scholar

[46]

F. Treves, An abstract nonlinear Cauchy-Kovalevska theorem, Trans. Amer. Math. Soc., 150 (1970), 77-92. 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, www.dm.ufscar.br/eventos/wpde2011. 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), 20140750, 20pp, http://dx.doi.org/10.1098/rspa.2014.0750. 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), 35-44, http://arxiv.org/pdf/1310.0268.pdf. 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), 248-276, http://arxiv.org/pdf/1301.3216v1 [nlin.SI]. 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), 1185-1196, arXiv:1205.2028v2. 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), 1411-1433. 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), 581-617. 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), 1113-1127. 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), 6644-6671. 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), 5153-5169. 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), 112-133. doi: 10.1016/j.na.2016.04.004.  Google Scholar

show all references

References:
[1]

S. Baouendi and C. Goulaouic, Remarks on the abstract form of nonlinear Cauchy-Kowalevski theorems, Comm. Partial Differential Equations, 2 (1977), 1151-1162. 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), 241-268. 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, 127 of the series Springer Proceedings in Mathematics & Statistics, (2015), 59-68. 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), 330-358. Google Scholar

[5]

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.  Google Scholar

[6]

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.  Google Scholar

[7]

A. Constantin, The Hamiltonian structure of the Camassa-Holm equation, Exposition. Math., 15 (1997), 53-85.  Google Scholar

[8]

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.  Google Scholar

[9]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. 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), 218-235. 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), 229-243. 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), 475-504. 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), 559-568.  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), 403-416. 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), 303-328.  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), 321-362. 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), 559-568. 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), 165-186. 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), 949-982. 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), 45-61. 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), 7129-7132. 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), 359-369. 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), 145-150. 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, Volume79, Cambridge University Press, 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), 827-839. 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), 575-584. 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), 2613-2625. 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), 016601, 13pp. 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), 63-82. 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), 27-63. 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), 29-39. 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, http://arxiv.org/abs/1507.05250, Jul. 2015. Google Scholar

[34]

L. Nirenberg, An abstract form of the nonlinear Cauchy-Kowalevski theorem, J. Differential Geom., 6 (1972), 561-576,  Google Scholar

[35]

T. Nishida, A note on a theorem of Nirenberg, J. Differential Geom., 12 (1977), 629-633. 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), 137-142.  Google Scholar

[37]

L. Ovsyannikov, A nonlinear Cauchy problems in a scale of Banach spaces, Dokl. Akad. Nauk. SSSR, 200 (1971), 789-792.  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), 1900-1906. 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), 309-341. 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), 112701, 9pp. 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), 082701, 20pp. 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), 1185-1196. 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), 309-327. 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) 013503 (9 pages). doi: 10.1063/1.3530865.  Google Scholar

[45]

F. Treves, Ovsyannikov theorem and hyperdifferential operators, Conselho Nacional de Pesquisas, Rio de Janeiro, 1968.  Google Scholar

[46]

F. Treves, An abstract nonlinear Cauchy-Kovalevska theorem, Trans. Amer. Math. Soc., 150 (1970), 77-92. 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, www.dm.ufscar.br/eventos/wpde2011. 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), 20140750, 20pp, http://dx.doi.org/10.1098/rspa.2014.0750. 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), 35-44, http://arxiv.org/pdf/1310.0268.pdf. 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), 248-276, http://arxiv.org/pdf/1301.3216v1 [nlin.SI]. 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), 1185-1196, arXiv:1205.2028v2. 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), 1411-1433. 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), 581-617. 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), 1113-1127. 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), 6644-6671. 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), 5153-5169. 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), 112-133. doi: 10.1016/j.na.2016.04.004.  Google Scholar

[1]

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