American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Heat equation with a nonlinear boundary condition and uniformly local $L^r$ spaces
  • DCDS Home
  • This Issue
  • Next Article
    Limiting distribution and error terms for the number of visits to balls in non-uniformly hyperbolic dynamical systems
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

Qiaoyi Hu 1, and  Zhijun Qiao 2,

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.
Keywords: weak kink solutions, unique continuation., persistence properties, Two-component system, peakon.
Mathematics Subject Classification: 35G25, 35L0.
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

[1]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081
[2]

Nicolas Rougerie. On two properties of the Fisher information. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020049
[3]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320
[4]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045
[5]

Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020467
[6]

Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020268
[7]

Mathew Gluck. Classification of solutions to a system of $ n^{\rm th} $ order equations on $ \mathbb R^n $. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5413-5436. doi: 10.3934/cpaa.2020246
[8]

Andreu Ferré Moragues. Properties of multicorrelation sequences and large returns under some ergodicity assumptions. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020386
[9]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248
[10]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340
[11]

Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073
[12]

Huu-Quang Nguyen, Ya-Chi Chu, Ruey-Lin Sheu. On the convexity for the range set of two quadratic functions. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020169
[13]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321
[14]

Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020273
[15]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049
[16]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253
[17]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138
[18]

Zhouchao Wei, Wei Zhang, Irene Moroz, Nikolay V. Kuznetsov. Codimension one and two bifurcations in Cattaneo-Christov heat flux model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020344
[19]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075
[20]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (49)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences