Differences between two methods to derive a nonlinear Schr?dinger equation and their application scopes

The present paper chooses a dusty plasma as an example to numerically and analytically study the differences between two different methods of obtaining nonlinear Schrodinger equation (NLSE). The first method is to derive a Korteweg-de Vries (KdV)-type equation and then derive the NLSE from the KdV-type equation, while the second one is to directly derive the NLSE from the original equation. It is found that the envelope waves from the two methods have different dispersion relations, different group velocities. The results indicate that two envelope wave solutions from two different methods are completely different. The results also show that the application scope of the envelope wave obtained from the second method is wider than that of the first one, though both methods are valuable in the range of their corresponding application scopes. It is suggested that, for other systems, both methods to derive NLSE may be correct, but their nonlinear wave solutions are different and their application scopes are also different.