Peaked wave solutions of Camassa–Holm equation☆

Explicit peaked wave solutions to the generalized Camassa-Holm equation

“…Our results generalize those in [3] and [10] and show that Conjecture 1 in [7] is incorrect. The method in this paper, different from bifurcation method of planar dynamical systems in [7] and [10], is simple and effective to find explicit solutions of nonlinear evolution equations.…”

“…In particular, we find new periodic cusp solutions expressed by the sine function for m = 1. When m = 2 or 3, we get new periodic cusp solutions to GCHE (1), which are expressed in terms of elliptic functions, and show that Conjecture 1 in [7] is incorrect. The method introduced in this paper is simple and effective to find explicit solutions of some nonlinear evolution equations.…”

“…When k = 0, m = 1 and a = 3, Eq. (1) is derived by Camassa and Holm [3] , and called the Camassa-Holm equation (CHE), see, for example, [2,7,10]. It is known that there exist peaked solitary wave and periodic cusp wave solutions to GCHE (1) for m = 1 [2,7,10−12] .…”

Approximate explicit solution of Camassa-Holm equation by He’s homotopy perturbation method

“…Fortunately, in recent years, a lot of new methods for solving nonlinear ordinary or partial differential equations are developed. For instance, Bäcklund transformation method [1], inverse scattering method [2], Darboux transformation method [3], Adomian's decomposition method [4], Hirota's bilinear method [5], homotopy analysis method [6,7], tanh-function method [8,9], sine-cosine method [10], the bifurcations method of planar dynamical systems [11,12], homotopy perturbation method (HPM) [13][14][15][16][17][18] etc. Generally speaking, all of the above methods have their own advantages and limitations respectively.…”

New travelling wave solutions to modified CH and DP equations

“…For the function u(x, t), the Camassa-Holm (CH) equation u t − u xxt + 3uu x = 2u x u xx + uu xxx (1) and the Degasperis-Procesi (DP) equation u t − u xxt + 4uu x = 3u x u xx + uu xxx (2) have been investigated by many researchers, for example, in [1][2][3][4][5][6][7][8][9][10]. The CH equation (1) is a shallow water equation and was originally derived as an approximation to the incompressible Euler equation, while the DP equation (2) can be considered as a model for shallow water dynamics.…”