Jacobian elliptic function method for nonlinear differential-difference equations

Discrete exact solutions to some nonlinear differential-difference equations via the (G′/G)-expansion method

“…We observe that the rational solutions (26) are presented here for the first time and they do not appear in [57][58][59].…”

“…, then we obtain u Ç n;3 ðtÞ ¼ Ç tanhðkÞ tanhðkn þ 2 sinðpÞ tanhðkÞt þ fÞ Â expðiðpn þ ð2 À 2 cosðpÞ sec h 2 ðkÞÞt þ dÞÞ; ð18Þ which are the known dark solitary wave solutions found by Dai and Zhang [57] and Dai et al [58] in which they are expressed as (20) and (34), respectively. However, in our case, we note that (18) is derived from a more general solution (16) in the sense that it contains more arbitrary parameters.…”

“…Likewise, assigning special values to the parameters in our solution (21), we can get many known solutions, for example, in Dai et al [57] as well as Huang and Liu [59]. We omit to display them for making the discussion short.…”

“…(9) was established by using the inverse scattering method [53], and has many interesting rich mathematical properties and physical applications including dynamics of an harmonic lattice [54], self-trapping on a dimer [4], and pulse dynamics in nonlinear optics [55]. The authors [56][57][58][59] successfully studied Eq. (9) by using the Exp-function and the Jacobian elliptic function expansion methods.…”

Exact Traveling Wave Solutions for Fitzhugh-Nagumo (FN) Equation and Modified Liouville Equation

“…In recent years, quite a few methods for obtaining explicit traveling and solitary wave solutions of nonlinear evolution equations have been proposed. A variety of powerful methods, tanh -sech method [1]- [3], extended tanh -method [4]- [6], sine -cosine method [7]- [9], homogeneous balance method [10,11],F-expansion method [12]- [14], exp-function method [15,16], trigonometric function series method [17], ( G G )− expansion method [18]- [21], Jacobi elliptic function method [22]- [25], The exp(−ϕ(ξ))-expansion method [26]- [28] and so on. The objective of this article is to apply The exp(−ϕ(ξ))-expansion method for finding the exact traveling wave solution of FitzhughNagumo (FN) equation and Modified Liouville equation which play an important role in biology and mathematical physics.…”