| Citation: |
Zhaqilao. MULTI-PEAKON SOLUTIONS TO A FOUR-COMPONENT CAMASSA-HOLM TYPE SYSTEM[J]. Journal of Applied Analysis & Computation, 2016, 6(4): 907-916. doi: 10.11948/2016059
|
MULTI-PEAKON SOLUTIONS TO A FOUR-COMPONENT CAMASSA-HOLM TYPE SYSTEM
-
Abstract
A four-component Camassa-Holm type system with cubic nonlinearity is investigated. It allows an arbitrary function Γ(x, t) to be involved in to include some existing integrable peakon equations as special reductions. We obtain N-peakon solutions of the four-component Camassa-Holm type system with two special cases of Γ(x, t). In particular, we give one-and two-peakon solutions in an explicit formula and are graphically plotted. Further, some interesting peaked solutions are found:some peakon waves possessing positive and negative amplitudes while others decaying and growing amplitudes. -
-
References
[1] M.S. Alber, R. Camassa, N.F. Yuri, D.D. Holm and J.E. Marsden, The complex geometry of weak piecewise smooth solutions of integrable nonlinear PDEs of shallow water and dym type, Commun. Math. Phys., 221(2001), 197-227. [2] R. Camassa and D.D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71(1993), 1661-1664. [3] A. Degasperis and M. Procesi, Asymptotic integrability symmetry and perturbation theory, in:A. Degasperis, G. Gaeta (Eds.), World Scientific, Singapore, 1999, 23-37. [4] A.S. Fokas, On a class of physically important integrable equations, Physica D, 87(1995), 145-150. [5] B. Fuchssteiner, Some tricks from the symmetry-toolbox for nonlinear equations:Generalizations of the Camassa-Holm equation, Physica D, 95(1996), 229-243. [6] X.G. Geng and B. Xue, A three-component generalization of Camassa-Holm equation with N-peakon solutions, Adv. Math., 226(2011), 827-839. [7] A.N.W. Hone and J.P. Wang, Integrable peakon equations with cubic nonlinearity, J. Phys. A:Math. Theor., 41(2008), 372002. [8] N.H. Li, Q.P. Liu and Z. Popowicz, A four-component Camassa-Holm type hierarchy, Journal of Geometry and Physics, 85(2014), 29-39. [9] P.J. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E, 53(1996), 1900-1906. [10] Z.J. Qiao, The Camassa-Holm hierarchy, N-dimensional integrable systems, and algebro-geometric solution on a symplectic submanifold, Commun. Math. Phys., 239(2003), 309-342. [11] Z.J. Qiao, Integrable hierarchy, 3×3 constrained systems, and parametric solutions, Acta Applicandae Mathematicae, 83(2004), 199-220. [12] Z.J. Qiao, A new integrable equation with cuspons and W/M-shape-peaks solitons, J. Math. Phys., 47(2006), 112701-09. [13] Z.J. Qiao, New integrable hierarchy, its parametric solutions, cuspons, one-peak solutions, and M/Wshape peak solitons, J. Math. Phys., 48(2007), 082701. [14] Z.J. Qiao and X.Q. Li, An integrable equation with nonsmooth solitons, Theor. Math. Phys., 167(2011), 584-589. [15] V. Novikov, Generalizations of the Camassa-Holm equation, J. Phys. A:Math. Theor., 42(2009), 342002. [16] J.F. Song, C.Z. Qu and Z.J. Qiao, A new integrable two-component system with cubic nonlinearity, J. Math. Phys., 52(2011), 013503. [17] B.Q. Xia and Z.J. Qiao, Integrable multi-component Camassa-Holm system, arXiv:1310.0268, Exactly Solvable and Integrable Systems (nlin.SI). [18] B.Q. Xia, Z.J. Qiao and R.G. Zhou, A synthetical integrable two-component model with peakon solutions, arXiv:1301.3216. -
-
Export File
Citation
Zhaqilao. MULTI-PEAKON SOLUTIONS TO A FOUR-COMPONENT CAMASSA-HOLM TYPE SYSTEM[J]. Journal of Applied Analysis & Computation, 2016, 6(4): 907-916. doi: 10.11948/2016059
DownLoad: