Exploring nonlinear drift waves: Limiting cases and dynamics

A general equation for drift waves is derived, incorporating both nonlinear electron density perturbation and ion vorticity effects. It is emphasized that the well-known Hasegawa–Mima (HM) equation for drift waves [A. Hasegawa and K. Mima, Phys. Fluids 21, 87 (1978)] includes only the ion vorticity term and neglects the nonlinear electron density contribution that naturally arises from the electron Boltzmann response. If the ion vorticity term is ignored, then the general nonlinear equation reduces to an equation that can give a two-dimensional soliton solution under an appropriate coordinate transformation. Furthermore, under the assumption that the normalized electrostatic potential depends only on one spatial coordinate along the predominant propagation direction, i.e., Φ = Φ(y), the equation reduces to a one-dimensional KdV equation [H. Saleem, Phys. Plasmas 31, 112102 (2024)]. Conversely, if the nonlinear electron density term is artificially suppressed and a two-dimensional potential Φ = Φ(x, y) is considered, the equation reduces to the Hasegawa–Mima equation supporting the dipolar vortex solution. Because the HM equation ignores the nonlinear electron density term, it cannot support one- or two-dimensional soliton solutions. Finally, the limiting forms of the general nonlinear equation are also briefly discussed using the reductive perturbation method.