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

Abstract

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.

Differences between two methods to derive a nonlinear Schrödinger equation and their application scopes

“…We now consider one-dimensional dust acoustic wave propagating in the x direction. The dimensionless equations of motion of the system can be given as follows: [77] ∂…”

“…It is found that A may attends to zero or A = 0 in certain regions of system parameters. [77] Generally, we can use the KdV equation to approximately describe nonlinear waves in dusty plasmas. However, when the coefficients in the KdV equation are zero, the equation is no longer applicable.…”

“…These coefficients can be tuned according to specific physical conditions, allowing the CKdV equation to more accurately describe the nonlinear wave in dusty plasmas, including phenomena that cannot be covered by the KdV equation. [77] Therefore, the CKdV equation can be obtained by using different perturbation methods: [74] and a CKdV equation [74] as follows:…”