Exact Solutions > Interesting Papers > N.A. Kudryashov. Seven common errors in finding exact solutions of nonlinear differential equations > 7. Sixth error... Seven common errors in finding exact solutions of nonlinear differential equations

 (7.1) 
 (7.2) 
 (7.3) 
However all these solutions are incorrect and do not satisfy Eq.(4.1). The solution of Eq.(4.1) can be found by many methods and can be presented in the form
 (7.4) 
where x_{0} is arbitrary constant.
Example 6b. Solution of the foam drainage equation by Bekir and Cevikel [66]
 (7.5) 
Eq.(7.5) was studied in [66] by means of the tanh  coth method by Bekir and Cevikel [66]. The authors proposed the method to obtain ”new travelling wave solutions”.
Using the travelling wave
 (7.6) 
they obtained the first  order equation in the form
 (7.7) 
and obtained three solutions
 (7.8) 
The solution of Eq.(7.7) can be found using the transformation . For function we obtain the Riccati equation in the form
 (7.9) 
Solution of Eq.(7.9) can be written as
 (7.10) 
We can see there are the misprints in solutions of Eq.(7.7)
Example 6c. ”Solutions” of the Fisher equation by Öziş and Köroǧlu [67]
 (7.11) 
Using the Expfunction method Öziş and Köroǧlu [67] found four ”solutions” of Eq. (7.11). These ”solutions” were given in the form
 (7.12) 
 (7.13) 
 (7.14) 
 (7.15) 
However all these expressions do not satisfy equation (7.11). We can note this fact without substituting solutions (7.12)  (7.15) into Eq. (7.11), because a true solution of Eq. (7.11) has the pole of the second order, but all functions (7.12)  (7.15) are the first order poles and certainly by substituting expressions (7.12)  (7.15) into equation (7.11) we do not obtain zero.
Example 6d. ”Solutions” of the modified Benjamin  Bona  Mahony equation (6.3) found by Yusufoglu [63].
The general solution of Eq.(6.4) can be found via the Jacobi elliptic function, but the author tried to obtain some solitary solutions by means of the Exp  function method using the travelling wave , . Some of his solutions are incorrect. For example the ”solution” (formula (3.12) [63])
 (7.16) 
do not satisfy Eq.(6.4).
The solution of Eq.(6.4) takes the form
 (7.17) 
This solution can be obtained by using different methods.
Example 6e. ”Solutions” of the Burgers  Huxley equation by Chun [68].
Chun [68] applied the Exp  function method to obtain the generalized solitary wave solutions of the Burgers — Huxley equation
 (7.18) 
In the travelling wave the author looked for the solution of equation
 (7.19) 
where . Some of the solutions by Chun are incorrect. In particular, ”solution” (25) in [68]
 (7.20) 
do not satisfy Eq.(7.19). The solutions of Eq.(7.18) were found in [69].
Example 6f. ”Solutions” of the Benjamin  Bona  Mahony  Burgers equation by ElWakil, Abdou and Hendi [70].
ElWakil and coauthors [70] using the Exp  function method looked for the solitary wave solutions of the BBMB equation
 (7.21) 
Taking the travelling wave the authors searched for the solution of the equation
 (7.22) 
Some of solutions by ElWakil and coauthors are incorrect. In particular, expression (18) in [70]
 (7.23) 
do not satisfy Eq.(7.22).
The solutions of Eq.(7.22) can be found taking the different methods into account. Note that integrating Eq.(7.22) with respect to we have
 (7.24) 
where C_{2} is an arbitrary constant. Eq.(7.24) coincides with the KdV  Burgers equation in the travelling wave. The solitary wave solutions of this equation are given above in section 1. The general solution of Eq.(7.24) can be found by analogue with the general solution of Eq.(2.7) [46].
Example 6g. ”Solutions” of the Klein — Gordon equation with quadratic nonlinearity by Zhang [71].
The Exp  function method was used by Zhang to obtain the generalized solitonary solutions of the Klein  Gordon equation with the quadratic nonlinearity
 (7.25) 
The author [71] applied the travelling wave and searched for the solution of the equation
 (7.26) 
At least two solutions of Eq.(7.26) by Zhang [71] are incorrect and do not satisfy Eq.(7.26)
 (7.27) 
 (7.28) 
The solitary wave solutions of Eq.(7.26) was obtained in many papers, because this equation coincides with KdV equation in the travelling wave [46]. These solutions can be written as the following
 (7.29) 
 (7.30) 
where is an arbitrary constant.
Example 6h. ”Solutions” of the Kuramoto  Sivashinsky equation by Noor, Mohyud  Din and Waheed [72].
Using the Exp  function method Noor, MohyudDin and Waheed [72] looked for the solutions of the Kuramoto—Sivashinsky equation
 (7.31) 
These authors presented two expressions as the solutions of the Kuramoto  Sivashinsky equation. Their first ”solution” is written as
 (7.32) 
where a_{0}, , k are arbitrary constants (formula (26) in [72]).
The second ”solution” by the authors takes the form
 (7.33) 
where a_{0}, a_{2}, a_{2}, b_{0}, b_{2}, , k are arbitrary constants (formula (34) in [72]).
Here we have the sixth common error in the fatal form. Substituting cited ”solutions” (7.32) and (7.33) into (7.31), we obtain that these ”solutions” do not satisfy the Kuramoto  Sivashinsky equation. Moreover, we cannot obtain zero with any nontrivial values of the parameters. We guess, that using the Exp  function method the authors [72] did not solve the system of the algebraic equations for the parameters.
The exact solutions of Eq.(7.31) were first found in [73]. These two solutions take the form
 (7.34) 
 (7.35) 
where C_{0} and x_{0} are arbitrary constants.
Many authors tried to find new exact solutions of the Kuramoto  Sivashinsky equation (7.31). Some of authors [36, 74, 75] believe that they found new solutions but it is not this case. Nobody cannot find new exact solutions of Eq.(7.31).
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