Generalized analytical solutions of a Korteweg–de Vries (KdV) equation with arbitrary real coefficients: Association with the plasma-fluid framework and physical interpretation

The Korteweg–de Vries (KdV) equation can be derived from a plasma-fluid model via a reductive perturbation technique. The associated methodology is summarized, from first principles, focusing on the underlying physical assumptions involved in the plasma-theoretical framework. A beam permeated electron-ion plasma is assumed, although the main findings of this study may be extended to more complicated plasma configurations. Rather counter-intuitively, it is shown that either of the (two) real coefficients appearing in the KdV equation (actually, both depending parametrically on the plasma configuration and on the beam characteristics) may take either positive or negative values, a possibility overlooked in the past. Different possibilities are investigated, from first principles, regarding the sign of the nonlinearity coefficient $A$ (that is determined by the electron background statistics, in combination with the beam velocity) and the sign of the dispersion coefficient $B$ (that is solely determined by the beam velocity and is always positive in its absence). The possibility of polarity reversal is investigated from first principles, in relation with both the electrostatic potential (pulse) profile and its associated electric field (bipolar pulse) $E=-\nabla \varphi$ in the electrostatic approximation. Different types of excitations are shown to exist and the role of the (sign of the) various coefficients in the pulse-shaped solution’s propagation characteristics is discussed.