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

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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.

Keywords: Peakon, Hamiltonian.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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.
[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.
[3]	S. C. Anco, in preparation.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[13]	A. Fokas, The Korteweg-de Vries equation and beyond, Acta Appl. Math., 39 (1995), 295-305. doi: 10.1007/BF00994638.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[23]	P. J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4350-2.
[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.

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References:

[1]	S. C. Anco and E. Recio, A general family of multi-peakon equations and their properties, arXiv: 1609.04354, math-ph.
[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.
[3]	S. C. Anco, in preparation.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[13]	A. Fokas, The Korteweg-de Vries equation and beyond, Acta Appl. Math., 39 (1995), 295-305. doi: 10.1007/BF00994638.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[23]	P. J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4350-2.
[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.

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