He's Semi-inverse Method for Camassa-Holm Equation and Simplied Modied Camassa-Holm Equation

Abstract

In this Letter, we study Camassa-Holm equation and Simplified Modified Camassa-Holm equation by using the well-known He's Semi-inverse Method. The solitary solutions are obtained using the Ritz method. In fact, the He's Semi-inverse Method is a promising method to various systems of linear and nonlinear equations.

New Traveling Wave Solutions to the Simplified Modified Camassa–Holm Equation and the Landau-Ginsburg-Higgs Equation

“…We contemplate the following simplified MCH model [16][17][18][19] : (7) This equation, which concerns the dispersion of water waves and has been widely studied, is particularly important in its field. It was originally developed to explain how shallow water waves propagate in a single direction over flat terrain.…”

“…Different equations have been studied using a variety of techniques. For the simplified form of modified Camassa-Holm (MCH) equation, Islam et al 16 applied the new auxiliary equation approach, Najafi et al 17 exploited the semi-inverse scattering method, and Alam and Akbar 18 investigated the same equation applying the generalized -expansion technique. Also, Liu et al 19 utilized the -expansion process to analyze the specified model.…”

A novel analytical technique to obtain the solitary solutions for nonlinear evolution equation of fractional order

“…Several authors have been discussed the simplified modified Camassa-Holm (SMCH) equation by using different techniques for finding exact traveling wave results. Particularly, Liu et al [22] used the (G /G)-expansion method, Najafi et al [23], used He's semi-inverse method applied exp(-ϕ(η))-expansion method and Redi et al [19] applied an improved (G /G)-expansion method; Gundogdu et al [28] applied the elliptic function expansion method to get the traveling wave solutions. Akber et al [26] obtained solutions u 1 (φ), u 3 (φ), u 6 (φ), u 7 (φ) and u 8 (φ) that are equivalent to our solutions u 10 (η), u 19 (η) and u 20 (η).…”

“…With the help of He's semi-inverse method [23] we have obtained the analytical solutions of the CH equation and the SMCH equation, Abbasbandy via the homotopy method [24] obtained traveling wave solutions of the SMCH equation. Recently many solitary solutions of SMCH equation were obtained [25,26].…”