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Rescaled expansivity and separating flows
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 |
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.
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] | |
[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] | |
[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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