A new method for finding exact traveling wave solutions to nonlinear partial differential equations

Exact traveling wave solutions to the (3+1)-dimensional mKdV–ZK and the (2+1)-dimensional Burgers equations via exp(−Φ(η))-expansion method

“…Many powerful methods to seek for exact solutions to the nonlinear partial differential equations have been proposed. Among these are the direct algebraic method [1], the Lie symmetry group method [2], the inverse scattering transform [3], the complex hyperbolic function method [4,5], the rank analysis method [6], the ansatz method [7][8][9][10][11][12][13][14][15][16][17][18], the ðG 0 =GÞ-expansion method [19][20][21][22][23][24][25][26], the modified simple equation method [27], the exp-functions method [28], the sine-cosine method [29], the Jacobi elliptic function expansion method [30,31], the F-expansion method [32], the Backlund transformation method [33], the Darboux transformation method [34], the homogeneous balance method [35][36][37], the Adomian decomposition method [38,39], the auxiliary parameter method [40], the homotopy perturbation method [41][42][43], the expðÀuðgÞÞ-expansion method [44]…”

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“…During the past four decades or so, the many researchers are interested to find powerful and efficient methods for analytic solutions of nonlinear equations. Many powerful methods to obtain exact solutions of nonlinear evolution equations have been constricted and developed such as the inverse scattering transform in [1], the Backlund/Darboux transform in [2][3][4], the Hirota's bilinear operators in [5], the truncated Painleve expansion in [6], the tanh-function expansion and its various extension in [7][8][9], the Jacobi elliptic function expansion in [10,11], the F-expansion in [12][13][14][15], the sub-ODE method in [16][17][18][19], the homogeneous balance method in [20][21][22], the sine-cosine method in [23,24] the rank analysis method in [25], the ansatz method in [26][27][28], the expfunction expansion method in [29] and so on, but there is no unified method that can be used to deal with all types of nonlinear evolution equations. )evolution equation with variable coefficients.…”

New Exact Solutions of Some Nonlinear Evolution Equations by Sub-Ode Method

“…Thus the methods for deriving exact solutions for the governing equations have to be developed. Many powerful methods are used to obtain travelling solitary wave solutions to nonlinear partial differential equations (PDEs) such as tanh method [12,13], the ansatz method [5,6], the sub-ODE method [3,4], Jacobi elliptic function method [1], exp-function method [7,8] and so on.…”