On travelling wave solutions of some nonlinear evolution equations

Abstract

A method to construct the exact solutions of some nonlinear evolution equations is presented by the hyperbolic function method. The efficiency of the method can be demonstrated by some nonlinear PDEs such as the BenjaminBona-Mahony equation, the coupled KdV and MKdV equations. New exact travelling wave solutions are presented. In addition, the properties of these equations are shown with figures.

Towards a Unified Method for Exact Solutions of Evolution Equations. An Application to Reaction Diffusion Equations with Finite Memory Transport

“…(2.2)-(2.4), we find that the tanh, exponential (function), ℘ (kz) and Jacobi-functions expansion methods presented in [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29] are equivalent to the mapping method. In the F -expansion method if one ignores the terms in F with negative powers, in the equation (1.4) one gets exactly the same results as those obtained by using the GMM, by bearing in mind that the second equation in (1.4) is identical to (2.2) when k = 2.…”

“…Their geometric structures varies from being solitary, soliton or doubly periodic waves. Seemingly, this motivated authors to present methods having solely the nomenclatures; tanh-function [15][16][17][18][19][20], Jacobi elliptic function expansion [21][22][23][24][25], Weierstrass elliptic functions [26,27] and theta elliptic function [28,29] expansion methods. These methods assume that the solution of (1.2) may be found as a polynomial of degree n in one of the specific functions mentioned in before.…”

TheG′G-expansion method for (2 + 1)-dimensional Kadomtsev–Petviashvili equation

“…In recent years, many powerful methods to construct exact solutions of nonlinear evolution equations have been established and developed such as the Jacobi elliptic function expansion, the tanh-method, the truncated Painleve expansion and the G 0 G À Á -expansion method (Fan, 2000;Inc and Evans, 2004;Liu et al, 2001;Yan, 2003;Yan and Zhang, 1999;Zayed et al, 2005;Zhang et al, 2008;Abdou, 2007;Malfliet, 1992;Parkes and Duffy, 1996;Wang and Li, 2005;Chow, 1995). The rest of the Letter is organized as follows.…”

Exact Traveling Wave Solutions for the (1 + 1)-Dimensional Compound KdVB Equation via the Novel (G'/G)-Expansion Method

“…On the other hand, not all equations posed by the advent of NLEEs models are readily solvable. As a result, many original techniques have been successfully urbanized by various groups of researchers, such as the Cole-Hopf transformation method [1], the Miura transformation method [2], the Hirota's bilinear method [3], the ( ) ( ) exp η −Φ -expansion method [4]- [6], the Sumudu transform method [7]- [14], the Fan sub-equation method [15] [16], the spectral-homotopy analysis method [17] [18], the least-squares finite element scheme [19], the (G′/G)-expansion method [20]- [23], the improved (G′/G)-expansion method [24], the trial function method [25], the nonlinear transform method [26], the extended Tanh-function method [27] [28], and the novel (G′/G)-expansion method [29]- [34], homotopy analysis method [35], to name a few. The latter sequence of papers really constituted a ladder honed in the current wealth of repeated experimental and theoretical successes that sprang us to the work at hand, that we hope will greatly benefit the readership, towards the further understanding of NLEEs dynamics and solutions, and mechanisms for recognizing and classifying them.…”