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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

Stephen Anco 1,, and  Daniel Kraus 2,

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 and Continuous Dynamical Systems, 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.

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.

[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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