# American Institute of Mathematical Sciences

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

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

 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 $p \neq 0$.
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:
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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.   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.   Google Scholar [3] G. W. Bluman and S. C. Anco, Symmetry and Integration Methods for Differential Equations,, Springer, (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.   Google Scholar [5] R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661.   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.   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.   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.   Google Scholar [9] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Math., 181 (1998), 229.   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.   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.   Google Scholar [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.   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.   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.   Google Scholar [16] A. N. W. Hone and J. P. Wang, Prolongation algebras and Hamiltonian operators for peakon equations,, Inverse Problems, 19 (2003), 129.   Google Scholar [17] P. J. Olver, Applications of Lie Groups to Differential Equations,, Springer-Verlag, (1993).   Google Scholar
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