Exact travelling wave solutions for some nonlinear (N+1)-dimensional evolution equations

Abstract

Abstract.In this paper, we implement the tanh-coth function method to construct the travelling wave solutions for (N + 1)-dimensional nonlinear evolution equations. Mathematical subject classification: 35K58, 35C06, 35A25.

Solving unsteady Korteweg–de Vries equation and its two alternatives

“…Hence, to seek for a new method and the expansion of existing methods are always welcomed, because there exists not a single method that can solve all types of NLEEs. As a result, many powerful methods for finding these exact solutions have been presented by diverse group of mathematicians and physicists, such as the Kudryashov method , the tanh–coth function method , the Exp‐function method , the Homotopy perturbation method , the modified simple equation method , the ( G ′/ G )‐expansion method , the exp(−Φ( ξ ))‐expansion method , and many more effective methods .…”

Exact traveling wave solutions of an autonomous system via the enhanced (G′/G)-expansion method

“…Thus, the methods for deriving exact solutions for the governing equations have to be developed. As a result, numerous techniques of obtaining traveling wave solutions have been developed over last three decades, such as, the tanh-coth function method, [2,3] the Kudryashov method, [4] the Exp-function method, [5][6][7][8] the modified simple equation method, [9][10][11][12] the (G′/G)-expansion method, [13][14][15][16][17][18][19][20][21][22] the exp(−Φ(ξ))-expansion method, [23,24] the Darboux transformation method, [25] the Hirota method, and [26] the differential transform method. [27][28][29][30] From our point of view, all these methods have some merits and demerits with respect to the problem considered and there is no unified method that can be used to deal with all types of NLEEs.…”

A note on improved F -expansion method combined with Riccati equation applied to nonlinear evolution equations

“…As a result, there has been a great amount of activity aimed at finding methods for solving not only NLEEs but also more general types of ordinary and partial differential equations. A list of several of the more well-known methods includes the solitary wave ansätz [1], the first integral method [2,3], the functional variable method [4,5], the Exp-function method [6][7][8][9][10], the modified simple equation method [11][12][13], the tanh-coth function method [14,15], the Kudryashov method [16,17], the exp(-Φ(ξ ))-expansion method [18], the (G /G)-expansion method [19][20][21][22], the homotopy perturbation method [23][24][25][26][27], the multiple exp-function method [28,29], Bernoulli sub-ODE method [30][31][32], the homotopy analysis method [33,34], the variational iteration method [35] and the F-expansion method [36,37].…”