# American Institute of Mathematical Sciences

September  2018, 38(9): 4449-4465. doi: 10.3934/dcds.2018194

## Hamiltonian structure of peakons as weak solutions for the modified Camassa-Holm equation

 1 Department of Mathematics and Statistics, Brock University, St. Catharines, ON L2S3A1, Canada 2 Department of Mathematics, SUNY Oswego, Oswego, NY 13126, USA

* Corresponding author

Received  September 2017 Revised  January 2018 Published  June 2018

The modified Camassa-Holm (mCH) equation is a bi-Hamiltonian system possessing $N$-peakon weak solutions, for all $N≥ 1$, in the setting of an integral formulation which is used in analysis for studying local well-posedness, global existence, and wave breaking for non-peakon solutions. Unlike the original Camassa-Holm equation, the two Hamiltonians of the mCH equation do not reduce to conserved integrals (constants of motion) for $2$-peakon weak solutions. This perplexing situation is addressed here by finding an explicit conserved integral for $N$-peakon weak solutions for all $N≥ 2$. When $N$ is even, the conserved integral is shown to provide a Hamiltonian structure with the use of a natural Poisson bracket that arises from reduction of one of the Hamiltonian structures of the mCH equation. But when $N$ is odd, the Hamiltonian equations of motion arising from the conserved integral using this Poisson bracket are found to differ from the dynamical equations for the mCH $N$-peakon weak solutions. Moreover, the lack of conservation of the two Hamiltonians of the mCH equation when they are reduced to $2$-peakon weak solutions is shown to extend to $N$-peakon weak solutions for all $N≥ 2$. The connection between this loss of integrability structure and related work by Chang and Szmigielski on the Lax pair for the mCH equation is discussed.

Citation: Stephen Anco, Daniel Kraus. Hamiltonian structure of peakons as weak solutions for the modified Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4449-4465. doi: 10.3934/dcds.2018194
##### References:
 [1] S. C. Anco and E. Recio, A general family of multi-peakon equations and their properties, arXiv: 1609.04354, math-ph.Google Scholar [2] S. C. Anco, Peakons: Weak solutions or distributions?, Abstract of contributed paper at 10th IMACS International Conference on Nonlinear Evolution Equations and Wave Phenomena (Athens, USA), 2017.Google Scholar [3] S. C. Anco, in preparation.Google Scholar [4] X. Chang and J. Szmigielski, Lax integrability of the modified Camassa-Holm equation and the concept of peakons, J. Nonlinear Math. Phys., 23 (2016), 563-572. doi: 10.1080/14029251.2016.1248156. Google Scholar [5] X. Chang and J. Szmigielski, Liouville integrability of conservative peakons for a modified Camassa-Holm equation, J. Nonlinear Math. Phys., 24 (2017), 584-595. doi: 10.1080/14029251.2017.1375693. Google Scholar [6] X. Chang and J. Szmigielski, Lax integrability and the peakon problem for the modified Camassa-Holm equation, Comm. Math. Phys., 358 (2018), 295-341. doi: 10.1007/s00220-017-3076-6. Google Scholar [7] X. Chang and J. Szmigielski, An inverse problem for the modified Camassa-Holm equation and multi-point Padé approximants, arXiv: 1512.08303, math-ph.Google Scholar [8] A. Constantin and W. A. 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 [9] A. Constantin and L. Molinet, Global weak solutions for a shallow water wave equation, Comm. Math. Phys., 211 (2000), 45-61. doi: 10.1007/s002200050801. Google Scholar [10] J. Escher, Y. Liu and Z. Yin, Global weak solutions and blow-up structure for the Degasperis-Procesi equation, J. Funct. Anal., 241 (2006), 457-485. doi: 10.1016/j.jfa.2006.03.022. Google Scholar [11] 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 [12] R. Camassa, D. D. Holm and J. M. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. doi: 10.1016/S0065-2156(08)70254-0. Google Scholar [13] A. Fokas, The Korteweg-de Vries equation and beyond, Acta Appl. Math., 39 (1995), 295-305. doi: 10.1007/BF00994638. Google Scholar [14] A. S. 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 [15] B. Fuchssteiner, Some tricks from the symmetry-toolbox for nonlinear equations: Generalizations o f the Camassa-Holm equation, Phys. D, 95 (1996), 229-243. doi: 10.1016/0167-2789(96)00048-6. Google Scholar [16] B. Fuchssteiner and A. S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D, 4 (1981), 47-66. doi: 10.1016/0167-2789(81)90004-X. Google Scholar [17] G. Gui, Y. Liu and L. Tian, Global existence and blow-up phenomena for the peakon $b$-family of equations, Indiana Univ. Math. J., 57 (2008), 1209-1234. doi: 10.1512/iumj.2008.57.3213. Google Scholar [18] G. Gui, Y. Liu, P. J. Olver and C. Qu, Wave-breaking and peakons for a modified Camassa-Holm equation, Commun. Math. Phys., 319 (2013), 731-759. doi: 10.1007/s00220-012-1566-0. Google Scholar [19] D. D. Holm and A. N. W. Hone, A class of equations with peakon and pulson solutions, J. Nonl. Math. Phys., 12 (2005), 380-394. doi: 10.2991/jnmp.2005.12.s1.31. Google Scholar [20] Y. Liu, P. J. Olver, C. Qu and S. Zhang, On the blow-up of solutions to the integrable modified Camassa-Holm equation, Analysis Appl., 12 (2014), 355-368. doi: 10.1142/S0219530514500274. Google Scholar [21] X. Liu, Y. Liu, P. J. Olver and C. Qu, Orbital stability of peakons for a generalization of the modified Camassa-Holm equation, Nonlinearity, 27 (2014), 2297-2319. doi: 10.1088/0951-7715/27/9/2297. Google Scholar [22] P. J. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev., 53 (1996), 1900-1906. doi: 10.1103/PhysRevE.53.1900. Google Scholar [23] P. J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4350-2. Google Scholar [24] Z. Qiao, A new integrable equation with cuspons and W/M-shape-peaks solitons, J. Math. Phys., 47 (2006), 112701 (9 pp). doi: 10.1063/1.2365758. Google Scholar

show all references

##### References:
 [1] S. C. Anco and E. Recio, A general family of multi-peakon equations and their properties, arXiv: 1609.04354, math-ph.Google Scholar [2] S. C. Anco, Peakons: Weak solutions or distributions?, Abstract of contributed paper at 10th IMACS International Conference on Nonlinear Evolution Equations and Wave Phenomena (Athens, USA), 2017.Google Scholar [3] S. C. Anco, in preparation.Google Scholar [4] X. Chang and J. Szmigielski, Lax integrability of the modified Camassa-Holm equation and the concept of peakons, J. Nonlinear Math. Phys., 23 (2016), 563-572. doi: 10.1080/14029251.2016.1248156. Google Scholar [5] X. Chang and J. Szmigielski, Liouville integrability of conservative peakons for a modified Camassa-Holm equation, J. Nonlinear Math. Phys., 24 (2017), 584-595. doi: 10.1080/14029251.2017.1375693. Google Scholar [6] X. Chang and J. Szmigielski, Lax integrability and the peakon problem for the modified Camassa-Holm equation, Comm. Math. Phys., 358 (2018), 295-341. doi: 10.1007/s00220-017-3076-6. Google Scholar [7] X. Chang and J. Szmigielski, An inverse problem for the modified Camassa-Holm equation and multi-point Padé approximants, arXiv: 1512.08303, math-ph.Google Scholar [8] A. Constantin and W. A. 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 [9] A. Constantin and L. Molinet, Global weak solutions for a shallow water wave equation, Comm. Math. Phys., 211 (2000), 45-61. doi: 10.1007/s002200050801. Google Scholar [10] J. Escher, Y. Liu and Z. Yin, Global weak solutions and blow-up structure for the Degasperis-Procesi equation, J. Funct. Anal., 241 (2006), 457-485. doi: 10.1016/j.jfa.2006.03.022. Google Scholar [11] 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 [12] R. Camassa, D. D. Holm and J. M. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. doi: 10.1016/S0065-2156(08)70254-0. Google Scholar [13] A. Fokas, The Korteweg-de Vries equation and beyond, Acta Appl. Math., 39 (1995), 295-305. doi: 10.1007/BF00994638. Google Scholar [14] A. S. 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 [15] B. Fuchssteiner, Some tricks from the symmetry-toolbox for nonlinear equations: Generalizations o f the Camassa-Holm equation, Phys. D, 95 (1996), 229-243. doi: 10.1016/0167-2789(96)00048-6. Google Scholar [16] B. Fuchssteiner and A. S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D, 4 (1981), 47-66. doi: 10.1016/0167-2789(81)90004-X. Google Scholar [17] G. Gui, Y. Liu and L. Tian, Global existence and blow-up phenomena for the peakon $b$-family of equations, Indiana Univ. Math. J., 57 (2008), 1209-1234. doi: 10.1512/iumj.2008.57.3213. Google Scholar [18] G. Gui, Y. Liu, P. J. Olver and C. Qu, Wave-breaking and peakons for a modified Camassa-Holm equation, Commun. Math. Phys., 319 (2013), 731-759. doi: 10.1007/s00220-012-1566-0. Google Scholar [19] D. D. Holm and A. N. W. Hone, A class of equations with peakon and pulson solutions, J. Nonl. Math. Phys., 12 (2005), 380-394. doi: 10.2991/jnmp.2005.12.s1.31. Google Scholar [20] Y. Liu, P. J. Olver, C. Qu and S. Zhang, On the blow-up of solutions to the integrable modified Camassa-Holm equation, Analysis Appl., 12 (2014), 355-368. doi: 10.1142/S0219530514500274. Google Scholar [21] X. Liu, Y. Liu, P. J. Olver and C. Qu, Orbital stability of peakons for a generalization of the modified Camassa-Holm equation, Nonlinearity, 27 (2014), 2297-2319. doi: 10.1088/0951-7715/27/9/2297. Google Scholar [22] P. J. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev., 53 (1996), 1900-1906. doi: 10.1103/PhysRevE.53.1900. Google Scholar [23] P. J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4350-2. Google Scholar [24] Z. Qiao, A new integrable equation with cuspons and W/M-shape-peaks solitons, J. Math. Phys., 47 (2006), 112701 (9 pp). doi: 10.1063/1.2365758. Google Scholar
 [1] Andrew N. W. Hone, Michael V. Irle. On the non-integrability of the Popowicz peakon system. Conference Publications, 2009, 2009 (Special) : 359-366. doi: 10.3934/proc.2009.2009.359 [2] Stephen C. Anco, Elena Recio, María L. Gandarias, María S. Bruzón. A nonlinear generalization of the Camassa-Holm equation with peakon solutions. Conference Publications, 2015, 2015 (special) : 29-37. doi: 10.3934/proc.2015.0029 [3] Kai Yan, Zhijun Qiao, Yufeng Zhang. On a new two-component $b$-family peakon system with cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5415-5442. doi: 10.3934/dcds.2018239 [4] 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 - A, 2016, 36 (12) : 6975-7000. doi: 10.3934/dcds.2016103 [5] 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 [6] Yongsheng Mi, Boling Guo, Chunlai Mu. Well-posedness and blow-up scenario for a new integrable four-component system with peakon solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2171-2191. doi: 10.3934/dcds.2016.36.2171 [7] Xi Tu, Zhaoyang Yin. Local well-posedness and blow-up phenomena for a generalized Camassa-Holm equation with peakon solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2781-2801. doi: 10.3934/dcds.2016.36.2781 [8] Xiuting Li, Lei Zhang. The Cauchy problem and blow-up phenomena for a new integrable two-component peakon system with cubic nonlinearities. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3301-3325. doi: 10.3934/dcds.2017140 [9] Kenneth R. Meyer, Jesús F. Palacián, Patricia Yanguas. Normally stable hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1201-1214. doi: 10.3934/dcds.2013.33.1201 [10] Hassan Najafi Alishah, Pedro Duarte. Hamiltonian evolutionary games. Journal of Dynamics & Games, 2015, 2 (1) : 33-49. doi: 10.3934/jdg.2015.2.33 [11] Antonio Giorgilli. Unstable equilibria of Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 855-871. doi: 10.3934/dcds.2001.7.855 [12] Wei Luo, Zhaoyang Yin. Local well-posedness in the critical Besov space and persistence properties for a three-component Camassa-Holm system with N-peakon solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5047-5066. doi: 10.3934/dcds.2016019 [13] V. Barbu. Periodic solutions to unbounded Hamiltonian system. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 277-283. doi: 10.3934/dcds.1995.1.277 [14] P. Balseiro, M. de León, Juan Carlos Marrero, D. Martín de Diego. The ubiquity of the symplectic Hamiltonian equations in mechanics. Journal of Geometric Mechanics, 2009, 1 (1) : 1-34. doi: 10.3934/jgm.2009.1.1 [15] David Damanik, Anton Gorodetski. The spectrum of the weakly coupled Fibonacci Hamiltonian. Electronic Research Announcements, 2009, 16: 23-29. doi: 10.3934/era.2009.16.23 [16] Răzvan M. Tudoran, Anania Gîrban. On the Hamiltonian dynamics and geometry of the Rabinovich system. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 789-823. doi: 10.3934/dcdsb.2011.15.789 [17] Ely Kerman. On primes and period growth for Hamiltonian diffeomorphisms. Journal of Modern Dynamics, 2012, 6 (1) : 41-58. doi: 10.3934/jmd.2012.6.41 [18] Gideon Simpson, Michael I. Weinstein, Philip Rosenau. On a Hamiltonian PDE arising in magma dynamics. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 903-924. doi: 10.3934/dcdsb.2008.10.903 [19] Claudio A. Buzzi, Jeroen S.W. Lamb. Reversible Hamiltonian Liapunov center theorem. Discrete & Continuous Dynamical Systems - B, 2005, 5 (1) : 51-66. doi: 10.3934/dcdsb.2005.5.51 [20] Edward Hooton, Pavel Kravetc, Dmitrii Rachinskii, Qingwen Hu. Selective Pyragas control of Hamiltonian systems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2019-2034. doi: 10.3934/dcdss.2019130

2018 Impact Factor: 1.143