# American Institute of Mathematical Sciences

January  2018, 38(1): 329-341. doi: 10.3934/dcds.2018016

## Symmetry analysis, persistence properties and unique continuation for the cross-coupled Camassa-Holm system

 School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China

* Corresponding author: ysjmath@163.com

Received  May 2017 Revised  July 2017 Published  September 2017

In this paper, we study symmetry analysis, persistence properties and unique continuation for the cross-coupled Camassa-Holm system. Lie symmetry analysis and similarity reductions are performed, some invariant solutions are also discussed. Then prove that the strong solutions of the system maintain corresponding properties at infinity within its lifespan provided the initial data decay exponentially and algebraically, respectively. Furthermore, we show that the system exhibits unique continuation if the initial momentum $m_0$ and $n_0$ are positive.

Citation: Shaojie Yang, Tianzhou Xu. Symmetry analysis, persistence properties and unique continuation for the cross-coupled Camassa-Holm system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 329-341. doi: 10.3934/dcds.2018016
##### References:
 [1] G. W. Bluman, Symmetry and Integration Methods for Differential Equations Springer, New York, 2002. Google Scholar [2] 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 [3] A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Appl. Anal., 5 (2007), 1-27. Google Scholar [4] R. Camassa and D. 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 [5] A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. R. Soc. Lond. Ser. A Math. Phys. Sci, 457 (2001), 953-970. doi: 10.1098/rspa.2000.0701. Google Scholar [6] A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5. Google Scholar [7] A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc., 44 (2007), 423-431. doi: 10.1090/S0273-0979-07-01159-7. Google Scholar [8] 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 [9] A. Constantin, Particle trajectories in extreme Stokes waves, IMA J. Appl. Math., 77 (2012), 293-307. doi: 10.1093/imamat/hxs033. Google Scholar [10] 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 [11] A. Constantin and W. Strauss, Stability of the Camassa-Holm solitons, J. Differential Equations, 12 (2002), 415-422. doi: 10.1007/s00332-002-0517-x. Google Scholar [12] 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. Google Scholar [13] A. Constantin and R. I. 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 [14] A. Constantin, Finite propagation speed for the Camassa-Holm equation, Journal of Mathematical Physics. 46(2005)023506. J. Math. Phys. 46 (2005), 023506.Google Scholar [15] A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Rational. Mech. Anal., 192 (2009), 165-186. doi: 10.1007/s00205-008-0128-2. Google Scholar [16] C. J. Cotter, D. D. Holm, R. I. Ivanov and J. R. Percival, Waltzing peakons and compacton pairs in a cross-coupled Camassa-Holm equation J. Phys. A: Math. Theor. 44 (2011), 265205. doi: 10.1088/1751-8113/44/26/265205. Google Scholar [17] H. Dai, Model equations for nolinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mech., 127 (1998), 193-207. doi: 10.1007/BF01170373. Google Scholar [18] R. Danchin, A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988. Google Scholar [19] R. Danchin, A note on well-posedness for Camassa-Holm equation, J. Differential Equations, 192 (2003), 429-444. doi: 10.1016/S0022-0396(03)00096-2. Google Scholar [20] A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäklund transformation and hereditary symmetries, Physica D, 4 (1981), 47-66. doi: 10.1016/0167-2789(81)90004-X. Google Scholar [21] Y. Fu, Y. Liu and C. Qu, On the blow-up structure for the generalized periodic Camassa-Holm and Degasperis-Procesi equations, J. Funct. Anal., 262 (2012), 3125-3158. doi: 10.1016/j.jfa.2012.01.009. Google Scholar [22] C. Guan and Z. Yin, Global weak solutions for a two-component Camassa-Holm shallow water system, J. Funct. Anal., 260 (2011), 1132-1154. doi: 10.1016/j.jfa.2010.11.015. Google Scholar [23] C. Guan and Z. Yin, Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system, J. Differential Equations, 248 (2010), 2003-2014. doi: 10.1016/j.jde.2009.08.002. Google Scholar [24] G. Gui and Y. Liu, On the global existence and wave-breaking criteria for the two-component Camassa-Holm system, J. Funct. Anal., 258 (2010), 4251-4278. Google Scholar [25] D. Henry, D. Holm and R. Ivanov, On the persistence properties of the cross-coupled Camassa-Holm system, J. Geom. Symmetry Phys., 32 (2013), 1-13. Google Scholar [26] D. Henry, Infinite propagation speed for a two component Camassa-Holm equation, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 597-606. doi: 10.3934/dcdsb.2009.12.597. Google Scholar [27] D. Henry, Compactly supported solutions of the Camassa-Holm equation, J. Nonlinear Math. Phys., 12 (2005), 342-347. doi: 10.2991/jnmp.2005.12.3.3. Google Scholar [28] D. Henry, Infinite propagation speed for the Degasperis-Procesi equation, J. Math. Anal. Appl., 311 (2005), 755-759. doi: 10.1016/j.jmaa.2005.03.001. Google Scholar [29] D. Henry, Compactly supported solutions of a family of nonlinear partial differential equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 15 (2008), 145-150. Google Scholar [30] D. Henry, Persistence properties for the Degasperis-Procesi equation, J. Hyperbolic Differ. Equ., 5 (2008), 99-111. doi: 10.1142/S0219891608001404. Google Scholar [31] D. Henry, Persistence properties for a family of nonlinear partial differential equations, Nonlinear Anal., 70 (2009), 1565-1573. doi: 10.1016/j.na.2008.02.104. Google Scholar [32] A. Himonas, C. Kenig and G. Misiolek, Non-uniform dependence for the periodic Camassa-Holm equation, Comm. Partial Differential Equations, 35 (2010), 1145-1162. doi: 10.1080/03605300903436746. Google Scholar [33] A. Himonas, G. Misiolek, G. Ponce and Y. Zhou, Persistence properties and unique continuation of solutions of the Camassa-Holm equation, Comm. Math. Phys., 271 (2007), 511-522. doi: 10.1007/s00220-006-0172-4. Google Scholar [34] H. Holden and X. Raynaud, Global conservative solutions of the Camassa-Holm equations-a Lagrangian point of view, Comm. Partial Differential Equations, 32 (2007), 1511-1549. doi: 10.1080/03605300601088674. Google Scholar [35] J. Li and Z. Yin, Remarks on the well-posedness of Camassa-Holm type equations in Besov spaces, J. Differential Equations, 261 (2016), 6125-6143. doi: 10.1016/j.jde.2016.08.031. Google Scholar [36] X. Liu, On the solutions of the cross-coupled Camassa-Holm system, Nonlinear Analysis: Real World Applications, 23 (2015), 183-195. doi: 10.1016/j.nonrwa.2014.12.004. Google Scholar [37] T. Lyons, Particle trajectories in extreme Stokes waves over infinite depth, Discrete Contin. Dyn. Syst., 34 (2014), 3095-3107. doi: 10.3934/dcds.2014.34.3095. Google Scholar [38] O. G. Mustafa, A note on the Degasperis-Procesi equation, J. Nonlinear Math. Phys., 12 (2005), 10-14. doi: 10.2991/jnmp.2005.12.1.2. Google Scholar [39] L. Ni and Y. Zhou, A new asymptotic behavior of solutions to the Camassa-Holm equation, Proc. Amer. Math. Soc., 140 (2012), 607-614. doi: 10.1090/S0002-9939-2011-10922-5. Google Scholar [40] P. Olver, Applications of Lie Groups to Differential Equations Springer, New York, 1986. doi: 10.1007/978-1-4684-0274-2. Google Scholar [41] L. V. Ovsiannikov, Group Analysis of Differential Equations Academic Press, 1982. Google Scholar [42] S. Zhou, Well-posedness and blowup phenomena for a cross-coupled Camassa-Holm equation with waltzing peakons and compacton pairs, J. Evol. Equ., 14 (2014), 727-747. doi: 10.1007/s00028-014-0236-4. Google Scholar [43] Y. Zhu and F. Fu, Persistence properties of the solutions to a generalized two-component Camassa-Holm shallow water system, Nonlinear Analysis: Theory, Methods and Applications, 128 (2015), 77-85. doi: 10.1016/j.na.2015.07.027. Google Scholar

show all references

##### References:
 [1] G. W. Bluman, Symmetry and Integration Methods for Differential Equations Springer, New York, 2002. Google Scholar [2] 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 [3] A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Appl. Anal., 5 (2007), 1-27. Google Scholar [4] R. Camassa and D. 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 [5] A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. R. Soc. Lond. Ser. A Math. Phys. Sci, 457 (2001), 953-970. doi: 10.1098/rspa.2000.0701. Google Scholar [6] A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5. Google Scholar [7] A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc., 44 (2007), 423-431. doi: 10.1090/S0273-0979-07-01159-7. Google Scholar [8] 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 [9] A. Constantin, Particle trajectories in extreme Stokes waves, IMA J. Appl. Math., 77 (2012), 293-307. doi: 10.1093/imamat/hxs033. Google Scholar [10] 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 [11] A. Constantin and W. Strauss, Stability of the Camassa-Holm solitons, J. Differential Equations, 12 (2002), 415-422. doi: 10.1007/s00332-002-0517-x. Google Scholar [12] 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. Google Scholar [13] A. Constantin and R. I. 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 [14] A. Constantin, Finite propagation speed for the Camassa-Holm equation, Journal of Mathematical Physics. 46(2005)023506. J. Math. Phys. 46 (2005), 023506.Google Scholar [15] A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Rational. Mech. Anal., 192 (2009), 165-186. doi: 10.1007/s00205-008-0128-2. Google Scholar [16] C. J. Cotter, D. D. Holm, R. I. Ivanov and J. R. Percival, Waltzing peakons and compacton pairs in a cross-coupled Camassa-Holm equation J. Phys. A: Math. Theor. 44 (2011), 265205. doi: 10.1088/1751-8113/44/26/265205. Google Scholar [17] H. Dai, Model equations for nolinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mech., 127 (1998), 193-207. doi: 10.1007/BF01170373. Google Scholar [18] R. Danchin, A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988. Google Scholar [19] R. Danchin, A note on well-posedness for Camassa-Holm equation, J. Differential Equations, 192 (2003), 429-444. doi: 10.1016/S0022-0396(03)00096-2. Google Scholar [20] A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäklund transformation and hereditary symmetries, Physica D, 4 (1981), 47-66. doi: 10.1016/0167-2789(81)90004-X. Google Scholar [21] Y. Fu, Y. Liu and C. Qu, On the blow-up structure for the generalized periodic Camassa-Holm and Degasperis-Procesi equations, J. Funct. Anal., 262 (2012), 3125-3158. doi: 10.1016/j.jfa.2012.01.009. Google Scholar [22] C. Guan and Z. Yin, Global weak solutions for a two-component Camassa-Holm shallow water system, J. Funct. Anal., 260 (2011), 1132-1154. doi: 10.1016/j.jfa.2010.11.015. Google Scholar [23] C. Guan and Z. Yin, Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system, J. Differential Equations, 248 (2010), 2003-2014. doi: 10.1016/j.jde.2009.08.002. Google Scholar [24] G. Gui and Y. Liu, On the global existence and wave-breaking criteria for the two-component Camassa-Holm system, J. Funct. Anal., 258 (2010), 4251-4278. Google Scholar [25] D. Henry, D. Holm and R. Ivanov, On the persistence properties of the cross-coupled Camassa-Holm system, J. Geom. Symmetry Phys., 32 (2013), 1-13. Google Scholar [26] D. Henry, Infinite propagation speed for a two component Camassa-Holm equation, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 597-606. doi: 10.3934/dcdsb.2009.12.597. Google Scholar [27] D. Henry, Compactly supported solutions of the Camassa-Holm equation, J. Nonlinear Math. Phys., 12 (2005), 342-347. doi: 10.2991/jnmp.2005.12.3.3. Google Scholar [28] D. Henry, Infinite propagation speed for the Degasperis-Procesi equation, J. Math. Anal. Appl., 311 (2005), 755-759. doi: 10.1016/j.jmaa.2005.03.001. Google Scholar [29] D. Henry, Compactly supported solutions of a family of nonlinear partial differential equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 15 (2008), 145-150. Google Scholar [30] D. Henry, Persistence properties for the Degasperis-Procesi equation, J. Hyperbolic Differ. Equ., 5 (2008), 99-111. doi: 10.1142/S0219891608001404. Google Scholar [31] D. Henry, Persistence properties for a family of nonlinear partial differential equations, Nonlinear Anal., 70 (2009), 1565-1573. doi: 10.1016/j.na.2008.02.104. Google Scholar [32] A. Himonas, C. Kenig and G. Misiolek, Non-uniform dependence for the periodic Camassa-Holm equation, Comm. Partial Differential Equations, 35 (2010), 1145-1162. doi: 10.1080/03605300903436746. Google Scholar [33] A. Himonas, G. Misiolek, G. Ponce and Y. Zhou, Persistence properties and unique continuation of solutions of the Camassa-Holm equation, Comm. Math. Phys., 271 (2007), 511-522. doi: 10.1007/s00220-006-0172-4. Google Scholar [34] H. Holden and X. Raynaud, Global conservative solutions of the Camassa-Holm equations-a Lagrangian point of view, Comm. Partial Differential Equations, 32 (2007), 1511-1549. doi: 10.1080/03605300601088674. Google Scholar [35] J. Li and Z. Yin, Remarks on the well-posedness of Camassa-Holm type equations in Besov spaces, J. Differential Equations, 261 (2016), 6125-6143. doi: 10.1016/j.jde.2016.08.031. Google Scholar [36] X. Liu, On the solutions of the cross-coupled Camassa-Holm system, Nonlinear Analysis: Real World Applications, 23 (2015), 183-195. doi: 10.1016/j.nonrwa.2014.12.004. Google Scholar [37] T. Lyons, Particle trajectories in extreme Stokes waves over infinite depth, Discrete Contin. Dyn. Syst., 34 (2014), 3095-3107. doi: 10.3934/dcds.2014.34.3095. Google Scholar [38] O. G. Mustafa, A note on the Degasperis-Procesi equation, J. Nonlinear Math. Phys., 12 (2005), 10-14. doi: 10.2991/jnmp.2005.12.1.2. Google Scholar [39] L. Ni and Y. Zhou, A new asymptotic behavior of solutions to the Camassa-Holm equation, Proc. Amer. Math. Soc., 140 (2012), 607-614. doi: 10.1090/S0002-9939-2011-10922-5. Google Scholar [40] P. Olver, Applications of Lie Groups to Differential Equations Springer, New York, 1986. doi: 10.1007/978-1-4684-0274-2. Google Scholar [41] L. V. Ovsiannikov, Group Analysis of Differential Equations Academic Press, 1982. Google Scholar [42] S. Zhou, Well-posedness and blowup phenomena for a cross-coupled Camassa-Holm equation with waltzing peakons and compacton pairs, J. Evol. Equ., 14 (2014), 727-747. doi: 10.1007/s00028-014-0236-4. Google Scholar [43] Y. Zhu and F. Fu, Persistence properties of the solutions to a generalized two-component Camassa-Holm shallow water system, Nonlinear Analysis: Theory, Methods and Applications, 128 (2015), 77-85. doi: 10.1016/j.na.2015.07.027. Google Scholar
The commutation table of Lie algebra
 $[V_i,V_j]$ $V_1$ $V_2$ $V_3$ $V_1$ 0 $-V_2$ 0 $V_2$ $V_2$ 0 0 $V_3$ 0 $0$ 0
 $[V_i,V_j]$ $V_1$ $V_2$ $V_3$ $V_1$ 0 $-V_2$ 0 $V_2$ $V_2$ 0 0 $V_3$ 0 $0$ 0
The adjoint representation
 $Ad(\exp(\epsilon V_i))V_j$ $V_1$ $V_2$ $V_3$ $V_1$ $V_1$ $e^\epsilon V_2$ $V_3$ $V_2$ $V_1-\epsilon V_2$ $V_2$ $V_3$ $V_3$ $V_1$ $V_2$ $V_3$
 $Ad(\exp(\epsilon V_i))V_j$ $V_1$ $V_2$ $V_3$ $V_1$ $V_1$ $e^\epsilon V_2$ $V_3$ $V_2$ $V_1-\epsilon V_2$ $V_2$ $V_3$ $V_3$ $V_1$ $V_2$ $V_3$
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