Article Contents
Article Contents

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

• 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 $p \neq 0$.
Mathematics Subject Classification: Primary: 34C14, 37J15; Secondary: 70S10.

 Citation:

•  [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. [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. [3] G. W. Bluman and S. C. Anco, Symmetry and Integration Methods for Differential Equations, Springer, New York, 2002. [4] R. Camassa, D. D. Holm and J. M. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. [5] R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. [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. [7] A. F. Cheviakov, GeM software package for computation of symmetries and conservation laws of differential equations, Comp. Phys. Comm., 176 (2007), 48-61. [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. [9] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. [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. [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. [12] A. Degasperis, D. D. Holm, A. N. W. Hone, A new integrable equation with peakon solutions, Theor. Math. Phys. 133 (2002), 1463-1474. [13] M. Fisher and J. Schiff, The Camassa Holm equation: conserved quantities and the initial value problem, Phys. Lett. A, 259 (1999), 371-376. [14] B. Fuchssteiner and A. S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Physica D, 4 (1981/1982), 47-66. [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. [16] A. N. W. Hone and J. P. Wang, Prolongation algebras and Hamiltonian operators for peakon equations, Inverse Problems, 19 (2003), 129-45. [17] P. J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 1993.
Open Access Under a Creative Commons license