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

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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

Abstract
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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.
[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.
[3]	G. W. Bluman and S. C. Anco, Symmetry and Integration Methods for Differential Equations,, Springer, (2002).
[4]	R. Camassa, D. D. Holm and J. M. Hyman, A new integrable shallow water equation,, Adv. Appl. Mech., 31 (1994), 1.
[5]	R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661.
[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.
[7]	A. F. Cheviakov, GeM software package for computation of symmetries and conservation laws of differential equations,, Comp. Phys. Comm., 176 (2007), 48.
[8]	A. Constantin, Existence of permanent and breaking waves for a shallow water equation: a geometric approach,, Ans. Inst. Fourier (Grenoble), 50 (2000), 321.
[9]	A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Math., 181 (1998), 229.
[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.
[11]	A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation,, Ann. Scuola Norm. Sup. Pisa, 26 (1998), 303.
[12]	A. Degasperis, D. D. Holm, A. N. W. Hone, A new integrable equation with peakon solutions,, Theor. Math. Phys. 133 (2002), 133 (2002), 1463.
[13]	M. Fisher and J. Schiff, The Camassa Holm equation: conserved quantities and the initial value problem,, Phys. Lett. A, 259 (1999), 371.
[14]	B. Fuchssteiner and A. S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries,, Physica D, 4 (): 47.
[15]	D. D. Holm, A. N. W. Hone, A class of equations with peakon and pulson solutions,, J. Nonlinear Math. Phys., 12 (2005), 380.
[16]	A. N. W. Hone and J. P. Wang, Prolongation algebras and Hamiltonian operators for peakon equations,, Inverse Problems, 19 (2003), 129.
[17]	P. J. Olver, Applications of Lie Groups to Differential Equations,, Springer-Verlag, (1993).

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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.
[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.
[3]	G. W. Bluman and S. C. Anco, Symmetry and Integration Methods for Differential Equations,, Springer, (2002).
[4]	R. Camassa, D. D. Holm and J. M. Hyman, A new integrable shallow water equation,, Adv. Appl. Mech., 31 (1994), 1.
[5]	R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661.
[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.
[7]	A. F. Cheviakov, GeM software package for computation of symmetries and conservation laws of differential equations,, Comp. Phys. Comm., 176 (2007), 48.
[8]	A. Constantin, Existence of permanent and breaking waves for a shallow water equation: a geometric approach,, Ans. Inst. Fourier (Grenoble), 50 (2000), 321.
[9]	A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Math., 181 (1998), 229.
[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.
[11]	A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation,, Ann. Scuola Norm. Sup. Pisa, 26 (1998), 303.
[12]	A. Degasperis, D. D. Holm, A. N. W. Hone, A new integrable equation with peakon solutions,, Theor. Math. Phys. 133 (2002), 133 (2002), 1463.
[13]	M. Fisher and J. Schiff, The Camassa Holm equation: conserved quantities and the initial value problem,, Phys. Lett. A, 259 (1999), 371.
[14]	B. Fuchssteiner and A. S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries,, Physica D, 4 (): 47.
[15]	D. D. Holm, A. N. W. Hone, A class of equations with peakon and pulson solutions,, J. Nonlinear Math. Phys., 12 (2005), 380.
[16]	A. N. W. Hone and J. P. Wang, Prolongation algebras and Hamiltonian operators for peakon equations,, Inverse Problems, 19 (2003), 129.
[17]	P. J. Olver, Applications of Lie Groups to Differential Equations,, Springer-Verlag, (1993).

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