Integrable nonlinear PDEs as evolution equations derived from multi-ion fluid plasma models

Abstract

Various types of nonlinear partial-differential equations (PDEs) have been proposed in relation with plasma dynamics. In a 1D geometry, a Korteweg–de Vries (KdV) equation can be derived from a plasma fluid model for electrostatic excitations, by means of the reductive perturbation technique proposed by Taniuti and coworkers in the 1960s (Washimi and Taniuti, 1966). In the simplest “textbook level” case of an electron–ion plasma, describing an ion fluid evolving against an electron distribution (assumed known), this integrable equation, incorporating a nonlinearity coefficient (say, $A$) and a dispersion coefficient ($B$), possesses analytical soliton solutions, whose polarity depends on the sign of $A$. In more elaborate plasma configurations, including a (one, or more) (negatively charged) secondary ion or/and electron population(s), a critical plasma composition where the quadratic nonlinearity term A is negligible may be possible: in this case, a modified KdV (mKdV) equation may be derived, where dispersion is balanced by a cubic nonlinearity term, leading to exact pulse-shaped soliton solutions. A third possible scenario occurs when, depending on the relative concentration between positive and negative ions in the plasma mixture, an extended KdV (i.e. a Gardner) equation may be obtained, allowing for both positive and negative soliton solutions.

In this study, we have revisited the reductive perturbation technique, using a generic bi-fluid (electronegative plasma) model as starting point, in an effort to elucidate the subtleties underlying the reduction of a fluid plasma model to a nonlinear evolution equation for the electrostatic (ES) potential. Considering different plasma compositions, different types of PDEs have been obtained, in specific regimes. We have thus studied the conditions for the existence of various types of pulse-shaped excitations (solitary waves) for the electrostatic potential, associated with bipolar electric field ($E=-\frac{\partial \varphi }{\partial x}$) waveforms, such as the ones observed in planetary magnetospheres and in laboratory experiments.