New exact solutions for the Kawahara equation using Exp-function method

Exact solutions of the Kawahara equation by Assas [L.M.B. Assas, J. Com. Appl. Math. 233 (2009) 97–102] are analyzed. It is shown that all solutions do not satisfy the Kawahara equation and consequently all nontrivial solutions by Assas are wrong. Recently Assas in [1] looked for exact solutions of the Kawahara equation using the Exp function method. This equation can be written as ut + α uux + β uxxx − γ uxxxxx = 0, (1) where α, β and γ are constants. The author [1] considered the Kawahara equation using the traveling wave z = k x + w t and tried to find new exact solutions of the nonlinear ordinary differential equation γ k Uzzzzz − β k Uzzz − α k U Uz − w Uz = 0. (2) He believe that he found a few exact solutions of Eq.(2) but this is not the case. It is obviously that the author [1] can not obtain any new nontrivial solutions of Eq.(2) using his approach. The matter is the general solution of Eq.(2) has the pole of the fourth order. This fact has an important bearing on the choose of expressions in the Exp-function method. However the author does not take this result into account. To look for exact solutions of the Kawahara equation the author [1] must take at least five terms in denominator and numerator of the ansatz in the Exp-function method. We can see that all constructions by author (formulae (15), (21) and (27) of work [1]) have poles of the second and the third order. Consequently we can expect that Assas can not find any solutions of the Kawahara equation. Just to be on the save side we have checked all solutions by Assas [1] and obtained that all solutions [1] do not satisfy the Kawahara equation except solution (17) that is the trivial solution u = a−1 b1 . Naturally solutions (18), (19) and (20) are trivial solutions as well because they were obtained from solution (17). We were surprised when we read that the author [1] has compared the accuracy his exact solutions with known exact solution and obtained that his results are ”very precise and that there is an inverse relationship between distance and