BGK Electron Solitary Waves Reexamined

This paper reexamines the physical roles of trapped and passing electrons in electron Bernstein-Greene-Kruskal (BGK) solitary waves, also called the BGK phase space electron holes (EH). It is shown that the charge density variation in the vicinity of the solitary potential is a net balance of the negative charge from trapped electrons and positive charge due to the decrease of the passing electron density. A BGK EH consists of electron density enhancements as well as a density depletion, instead of only the density depletion as previously thought. The shielding of the positive core is not a thermal screening by the ambient plasma, but achieved by trapped electrons oscillating inside the potential energy trough. The total charge of a BGK EH is therefore zero. Two separated EHs do not interact and the concept of negative mass is not needed. These features are independent of the strength of the nonlinearity. BGK EHs do not require thermal screening, and their size is thus not restricted to be greater than the Debye length $\lambda_D$. Our analysis predicts that BGK EHs smaller than $\lambda_D$ can exist. A width($\delta$)-amplitude($\psi$) relation of an inequality form is obtained for BGK EHs in general. For empty-centered EHs with potential amplitude $\gg 1$, we show that the width-amplitude relation of the form $\delta\propto\sqrt{\psi}$ is common to bell-shaped potentials. For $\psi\ll 1$, the width approaches zero faster than $\sqrt{\psi}$.