The binary F-expansion method and its application for solving the (n+1)-dimensional sine-Gordon equation

Numerical simulation of two-dimensional sine-Gordon solitons via a local weak meshless technique based on the radial point interpolation method (RPIM)

“…Physical applications of solitons have been found among others in shallow-water waves, optical fibers, Josephson-junction oscillators, etc. We also refer the interested reader to [48,51] for more applications.…”

New exact solutions for some power law nonlinear diffusion equation

“…There are no Weierstrass elliptic solutions to (12) consequent of ansatz (11) as the relation C 1 = b 2c , of solution set (14), is at variance with such.…”

“…In effect, results obtained herein can be applied directly to sub-classes of (2). Recently, various auxiliary equation methods have been proposed for studying exact TW solutions of nonlinear partial differential equations [7][8][9][10][11][12][13][14]. The auxiliary equation of the modification presented here generalizes those of [12,13] and furthermore allows explicitly two types of TW solutions for certain sub-classes of (1), one which is dispersive and excludes the existence of elliptic TWs and the other which is not dispersive and permits elliptic TWs.…”

New periodic wave solutions for nonlinear evolution equations with variable coefficients via mapping method

“…As a result, the research on exact solutions of nonlinear evolution equations becomes more and more important, such as the famous Hirota's method [4], the Backlund and Darboux transformation [5][6][7], Painleve expansions [8], homogeneous balance method [9], Jacobi elliptic function [10,11], extended tanh-function methods [12][13][14], extended F-expansion methods [15][16][17][18][19], variational iteration methods [20][21][22][23], Adomian methods [24,25] and extended mapping method [26,27], Exp-function method [2] and which was proposed recently as an overall generalization of Jacobi elliptic expansion function method. Most of exact solutions were obtained by these methods, including the solitary wave solutions, shock wave solution, periodic wave solutions, and so on.…”