# 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 and Continuous Dynamical Systems, 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. [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. [3] A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Appl. Anal., 5 (2007), 1-27. [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. [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. [6] A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.  doi: 10.1007/s00222-006-0002-5. [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. [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. [9] A. Constantin, Particle trajectories in extreme Stokes waves, IMA J. Appl. Math., 77 (2012), 293-307.  doi: 10.1093/imamat/hxs033. [10] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math, 181 (1998), 229-243.  doi: 10.1007/BF02392586. [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. [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. [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. [14] A. Constantin, Finite propagation speed for the Camassa-Holm equation, Journal of Mathematical Physics. 46(2005)023506. J. Math. Phys. 46 (2005), 023506. [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. [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. [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. [18] R. Danchin, A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [30] D. Henry, Persistence properties for the Degasperis-Procesi equation, J. Hyperbolic Differ. Equ., 5 (2008), 99-111.  doi: 10.1142/S0219891608001404. [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. [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. [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. [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. [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. [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. [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. [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. [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. [40] P. Olver, Applications of Lie Groups to Differential Equations Springer, New York, 1986. doi: 10.1007/978-1-4684-0274-2. [41] L. V. Ovsiannikov, Group Analysis of Differential Equations Academic Press, 1982. [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. [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.

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##### References:
 [1] G. W. Bluman, Symmetry and Integration Methods for Differential Equations Springer, New York, 2002. [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. [3] A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Appl. Anal., 5 (2007), 1-27. [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. [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. [6] A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.  doi: 10.1007/s00222-006-0002-5. [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. [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. [9] A. Constantin, Particle trajectories in extreme Stokes waves, IMA J. Appl. Math., 77 (2012), 293-307.  doi: 10.1093/imamat/hxs033. [10] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math, 181 (1998), 229-243.  doi: 10.1007/BF02392586. [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. [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. [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. [14] A. Constantin, Finite propagation speed for the Camassa-Holm equation, Journal of Mathematical Physics. 46(2005)023506. J. Math. Phys. 46 (2005), 023506. [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. [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. [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. [18] R. Danchin, A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [30] D. Henry, Persistence properties for the Degasperis-Procesi equation, J. Hyperbolic Differ. Equ., 5 (2008), 99-111.  doi: 10.1142/S0219891608001404. [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. [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. [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. [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. [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. [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. [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. [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. [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. [40] P. Olver, Applications of Lie Groups to Differential Equations Springer, New York, 1986. doi: 10.1007/978-1-4684-0274-2. [41] L. V. Ovsiannikov, Group Analysis of Differential Equations Academic Press, 1982. [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. [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.
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
 $Ad(\exp(\epsilon V_i))V_j$ $V_1$ $V_2$ $V_3$ $V_1$ $V_1$ $e^\epsilon V_2$ $V_3$ $V_2$ $V_1-\epsilon V_2$ $V_2$ $V_3$ $V_3$ $V_1$ $V_2$ $V_3$
 $Ad(\exp(\epsilon V_i))V_j$ $V_1$ $V_2$ $V_3$ $V_1$ $V_1$ $e^\epsilon V_2$ $V_3$ $V_2$ $V_1-\epsilon V_2$ $V_2$ $V_3$ $V_3$ $V_1$ $V_2$ $V_3$
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