Comment on: ”Multi soliton solution, rational solution of the Boussinesq-Burgers equations”

Abstract

We demonstrate that all ”new” exact solutions of the Boussinesq Burgers equations by Rady A.S.A., Osman, E.S., Khalfallah M., Communications in Nonlinear Science and Numerical Simulation (2009) doi:10.10016/j.cnsns.2009.05.053] are well known and were obtained many years ago. In work [1] Rady, Osman and Khalfallah have found multi soliton solution, rational solution and ”new trigonometric function periodic solutions” of the Boussinesq-Burgers equations ut + 2uux − 12 vx = 0, (1) vt + 2(uv)x − 12 uxxx = 0. (2) It is known [2] that the system of equations (1), (2) has the Lax pair and the Cauchy problem for this system can be solved by the inverse scattering transform. Nevertheless the authors [1] decided to consider this system again using the peculiar approach. They do not study the system of Eqs.(1),(2) but use the additional condition v = −ux. (3) Assuming Eq.(3) the authors believe that they still considered the system of equations (1) (2) but this is not the case. As result of the condition (3) is that the authors obtained the system of equations which is equivalent to the well known Burgers equation with negative viscosity [3, 4] ut + 2uux + 1 2 uxx = 0. (4) So Rady, Osman and Khalfallah in the work [1] studied the Burgers equation and presented all results for this equation. However the authors [1] did not give the name of this equation and they did not present references for many results corresponding to this equation. Eq.(4) was first studied by Batemann [3] but we know this equation in periodic literature as the Burgers equation [4]. There is the remarkable transformation that is the Cole-Hopf transformation for the Eq.(4) [5, 6] u = 1 2 ∂ ln F ∂x . (5)

Cite this paper

@inproceedings{Kudryashov2010CommentO, title={Comment on: ”Multi soliton solution, rational solution of the Boussinesq-Burgers equations”}, author={Nikolai A. Kudryashov and Mikhail B. Soukharev}, year={2010} }