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. EscherY. 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. CamassaD. 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. GuiY. 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. GuiY. LiuP. 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. LiuP. J. OlverC. 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. LiuY. LiuP. 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. EscherY. 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. CamassaD. 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. GuiY. 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. GuiY. LiuP. 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. LiuP. J. OlverC. 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. LiuY. LiuP. 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]

Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024

[2]

Simon Hochgerner. Symmetry actuated closed-loop Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 641-669. doi: 10.3934/jgm.2020030

[3]

Hua Shi, Xiang Zhang, Yuyan Zhang. Complex planar Hamiltonian systems: Linearization and dynamics. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020406

[4]

Manuel de León, Víctor M. Jiménez, Manuel Lainz. Contact Hamiltonian and Lagrangian systems with nonholonomic constraints. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2021001

[5]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[6]

Adrian Viorel, Cristian D. Alecsa, Titus O. Pinţa. Asymptotic analysis of a structure-preserving integrator for damped Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020407

[7]

Huanhuan Tian, Maoan Han. Limit cycle bifurcations of piecewise smooth near-Hamiltonian systems with a switching curve. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020368

[8]

Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Ground state homoclinic orbits for a class of asymptotically periodic second-order Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1627-1652. doi: 10.3934/dcdsb.2020176

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (121)
  • HTML views (121)
  • Cited by (4)

Other articles
by authors

[Back to Top]