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Limiting distribution and error terms for the number of visits to balls in non-uniformly hyperbolic dynamical systems
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 |
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. |
show all references
References:
[1] |
A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239.
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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