Application of reductive perturbation method in dusty plasma and its applicable scope

Perturbation methods are widely employed to study wave phenomena in nonlinear systems, typically assuming small amplitude and long wavelength approximations. However, the conditions for establishing these assumptions have not yet been fully resolved. This study focuses on dusty plasma systems, aiming to elucidate the principles of perturbation methods and to determine their scope of applicability through numerical simulations. For pulse-type nonlinear waves, the KdV equation accurately describes them when the amplitude is sufficiently small and the wavelength is sufficiently long. In dusty plasma systems, the NLSE accurately describes nonlinear waves when background waves are present, the amplitude is small, and the envelope wavelength is sufficiently long. The term “small amplitude" refers to that perturbations are much smaller than their equilibrium states. The “long wavelength approximation" indicates that the wavelength is significantly larger than the system's characteristic length (for the dusty plasma systems, this characteristic length is the Debye length). The methods and conclusions of this study are applicable not only to dusty plasma systems but also to other systems, offering valuable guidance and reference for applying perturbation methods across various fields.