Comment on “Electrostatic compressive and rarefactive shocks and solitons in relativistic plasmas occurring in polar regions of pulsar”

Abstract

The aim of this comment is to show the solution of the KdVB equation used by Shah et al. (Astrophys. Space Sci. 335:529–537, 2011, doi:10.1007/s10509-011-0766-y) is not correct. So, the numerical results that are predicted in this manuscript should not be helpful for further investigations in a plasma laboratory. For this reason, we have employed the Bernoulli’s equation method to obtain the correct form of analytical solution to this equation, which is appropriate for the study of electrostatic compressive and rarefactive shocks and solitons in relativistic plasmas occurring in polar regions of pulsar.

Appendix 1

We have cheeked our obtain solution to the KdVB equation with aid of Maple software as follows:

$$\begin{aligned} &{>} \zeta:= \xi - v \tau \\ & \begin{aligned} {>}\phi^{ ( 1 )}&:= \biggl( \frac{v}{A} - \frac{6 C^{2}}{25 BA} \tanh \biggl( \frac{C}{10 B}. \zeta \biggr) \biggr)\\ &{}\quad + \frac{3. C^{2}}{25. B. A} \biggl( 1 - \tanh^{2} \biggl( \frac{C}{10 B}. \zeta \biggr) \biggr) \end{aligned} \\ &{>} \mathit{diff}\bigl(\phi^{ ( 1 )}, \tau \bigr) \\ &{>}\mathit{diff}\bigl(\phi^{ ( 1 )}, \xi \bigr) \\ &{>} \mathit{diff}\bigl(\phi^{ ( 1 )}, \xi \$2\bigr) \\ & {>} \mathit{diff}\bigl( \phi^{ ( 1 )}, \xi \$3\bigr) \\ & \begin{aligned} {>} &\mathit{diff}\bigl(\phi^{ ( 1 )}, \tau \bigr)+A. \phi^{ ( 1 )}. \mathit{diff} \bigl(\phi^{ ( 1 )}, \xi \bigr)+B. \mathit{diff}\bigl( \phi^{ ( 1 )}, \xi \$3\bigr)\\ &{}\quad -C. \mathit{diff}\bigl(\phi^{ ( 1 )}, \xi \$2\bigr) \end{aligned} \\ &{>} \mathit{Simplify}(\%) \end{aligned}$$

The output is zero (0).

It is provided that the obtained solution (22) in this article satisfies the KdVB equation (6).