Application scope of the reductive perturbation method to derive the KdV equation and CKdV equation in dusty plasma

The application scopes of two different reductive perturbation methods to derive the Korteweg-de Vries (KdV) equation and coupled KdV (CKdV) equation in two-temperature-ion dusty plasma are given by using the particle-in-cell (PIC) numerical method in the present paper. It suggests that the reductive perturbation method (RPM) is valid if the amplitude of the CKdV solitary wave is small enough. However, for the KdV solitary wave, RPM is valid not only if the amplitude of the KdV solitary wave is small enough, but also if the nonlinear coefficient of the KdV equation is not tending to zero.