American Institute of Mathematical Sciences

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2015, 2015(special): 29-37. doi: 10.3934/proc.2015.0029

A nonlinear generalization of the Camassa-Holm equation with peakon solutions

Stephen C. Anco 1, , Elena Recio 1, , María L. Gandarias 2, and  María S. Bruzón 2,

1. 

Department of Mathematics and Statistics, Brock University, St. Catharines, Ontario, L2S 3A1, Canada, Canada
2. 

Department of Mathematics, Faculty of Sciences, University of Cádiz, Puerto Real, Cádiz 11510, Spain, Spain

Received  October 2014 Revised  March 2015 Published  November 2015

A nonlinearly generalized Camassa-Holm equation, depending an arbitrary nonlinearity power $p \neq 0$, is considered. This equation reduces to the Camassa-Holm equation when $p=1$ and shares one of the Hamiltonian structures of the Camassa-Holm equation. Two main results are obtained. A classification of point symmetries is presented and a peakon solution is derived, for all powers $p \neq 0$.
Keywords: travelling wave, Camassa-Holm equation, peakon, Hamiltonian., conservation laws, symmetries.
Mathematics Subject Classification: Primary: 34C14, 37J15; Secondary: 70S1.
Citation: 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
References:
[1]

M. S. Alber, R. Camassa, D. D. Holm and J. E. Marsden, The geometry of peaked solitons and billiard solutions of a class of integrable PDE's, Lett. Math. Phys., 32 (1994), 137-151. Google Scholar

[2]

S. C. Anco, P. L. da Silva, I. L. Freire, A family of wave-breaking equations generalizing the Camassa-Holm and Novikov equations, Lett. Math. Phys., 32 (1994), 137-151. Google Scholar

[3]

G. W. Bluman and S. C. Anco, Symmetry and Integration Methods for Differential Equations, Springer, New York, 2002. Google Scholar

[4]

R. Camassa, D. D. Holm and J. M. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. Google Scholar

[5]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. Google Scholar

[6]

C. S. Cao, D. D. Holm and E. S. Titi, Traveling wave solutions for a class of one-dimensional nonlinear shallow water wave models, J. Dyn. Diff. Eqs., 16 (2004), 167-178. Google Scholar

[7]

A. F. Cheviakov, GeM software package for computation of symmetries and conservation laws of differential equations, Comp. Phys. Comm., 176 (2007), 48-61. Google Scholar

[8]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: a geometric approach, Ans. Inst. Fourier (Grenoble), 50 (2000), 321-362. Google Scholar

[9]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. Google Scholar

[10]

A. Constantin and J. Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation, Math. Z., 33 (2000), 75-91. Google Scholar

[11]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa, 26 (1998), 303-328. Google Scholar

[12]

A. Degasperis, D. D. Holm, A. N. W. Hone, A new integrable equation with peakon solutions, Theor. Math. Phys. 133 (2002), 1463-1474. Google Scholar

[13]

M. Fisher and J. Schiff, The Camassa Holm equation: conserved quantities and the initial value problem, Phys. Lett. A, 259 (1999), 371-376. Google Scholar

[14]

B. Fuchssteiner and A. S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries,, Physica D, 4 (): 47.   Google Scholar

[15]

D. D. Holm, A. N. W. Hone, A class of equations with peakon and pulson solutions, J. Nonlinear Math. Phys., 12 (2005), 380-394. Google Scholar

[16]

A. N. W. Hone and J. P. Wang, Prolongation algebras and Hamiltonian operators for peakon equations, Inverse Problems, 19 (2003), 129-45. Google Scholar

[17]

P. J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 1993. Google Scholar

show all references

References:
[1]

M. S. Alber, R. Camassa, D. D. Holm and J. E. Marsden, The geometry of peaked solitons and billiard solutions of a class of integrable PDE's, Lett. Math. Phys., 32 (1994), 137-151. Google Scholar

[2]

S. C. Anco, P. L. da Silva, I. L. Freire, A family of wave-breaking equations generalizing the Camassa-Holm and Novikov equations, Lett. Math. Phys., 32 (1994), 137-151. Google Scholar

[3]

G. W. Bluman and S. C. Anco, Symmetry and Integration Methods for Differential Equations, Springer, New York, 2002. Google Scholar

[4]

R. Camassa, D. D. Holm and J. M. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. Google Scholar

[5]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. Google Scholar

[6]

C. S. Cao, D. D. Holm and E. S. Titi, Traveling wave solutions for a class of one-dimensional nonlinear shallow water wave models, J. Dyn. Diff. Eqs., 16 (2004), 167-178. Google Scholar

[7]

A. F. Cheviakov, GeM software package for computation of symmetries and conservation laws of differential equations, Comp. Phys. Comm., 176 (2007), 48-61. Google Scholar

[8]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: a geometric approach, Ans. Inst. Fourier (Grenoble), 50 (2000), 321-362. Google Scholar

[9]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. Google Scholar

[10]

A. Constantin and J. Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation, Math. Z., 33 (2000), 75-91. Google Scholar

[11]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa, 26 (1998), 303-328. Google Scholar

[12]

A. Degasperis, D. D. Holm, A. N. W. Hone, A new integrable equation with peakon solutions, Theor. Math. Phys. 133 (2002), 1463-1474. Google Scholar

[13]

M. Fisher and J. Schiff, The Camassa Holm equation: conserved quantities and the initial value problem, Phys. Lett. A, 259 (1999), 371-376. Google Scholar

[14]

B. Fuchssteiner and A. S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries,, Physica D, 4 (): 47.   Google Scholar

[15]

D. D. Holm, A. N. W. Hone, A class of equations with peakon and pulson solutions, J. Nonlinear Math. Phys., 12 (2005), 380-394. Google Scholar

[16]

A. N. W. Hone and J. P. Wang, Prolongation algebras and Hamiltonian operators for peakon equations, Inverse Problems, 19 (2003), 129-45. Google Scholar

[17]

P. J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 1993. Google Scholar

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