The ion-acoustic soliton: A gas-dynamic viewpoint

Abstract

The properties of fully nonlinear ion-acoustic solitons are investigated by interpreting conservation of total momentum as the structure equation for the proton flow in the wave. In most studies momentum conservation is regarded as the first integral of the Poisson equation for the electric potential and is interpreted as being analogous to a particle moving in a pseudo-potential well. By adopting an essentially gas-dynamic viewpoint, which emphasizes momentum conservation and the properties of the Bernoulli-type energy equations, the crucial role played by the proton sonic point becomes apparent. The relationship (implied by energy conservation) between the electron and proton speeds in the transition yields a locus—the hodograph of the system–which shows that, in the first half of the soliton, the electrons initially lag behind the protons until the charge neutral point is reached, after which they run ahead of the protons. The system reaches an equilibrium point (the center of the soliton) before the proton flow goes sonic. It follows that the critical ion-acoustic Mach number, Mc, above which smooth, continuous solitons cannot be constructed, stems from the requirement that the two equilibrium points of the structure equation coalesce at the proton sonic point of the flow. In general the range of the ion-acoustic Mach numbers, Mep, in which solitons exist, is extended beyond the classical range 1<Mep<1.6. In the special case of cold protons and hot electrons with an adiabatic index 2, the structure equation may be integrated in closed form. This analytic solution describes the fully nonlinear counterpart to the sech2 shaped pulses characteristic of weakly nonlinear waves and shows that solitons exist only if 1<Mep<2. The corresponding maximum potential, associated with the critical ion-acoustic Mach number, can be between 1.3kTe and 10kTe depending upon the values of the adiabatic indices of the electrons and protons and the proton Mach number.

Solitary waves in self-gravitating molecular clouds

“…We follow a recent gasdynamic approach (McKenzie 2002;Verheest et al 2004;Cattaert et al 2005) and study nonlinear structures in their own reference frame, emphasizing the gas-dynamical constraints on their existence. Such a description highlights the role of sonic and neutral points.…”

Effects of hot electron inertia on electron-acoustic solitons and double layers

“…4. It is seen that, for f increasing from 0 to 1, the soliton velocities are limited by, respectively, the cool electron sonic point 4,19 (green dotted-dashed curve), the hot electron sonic point (red solid curve), the occurrence of negative and positive double layers (blue dashed curve), and the model limitations (black dotted line). The model limit, where the Mach number equals the hot electron thermal speed, applies for f !…”

“…That may be when the soliton speed reaches the sonic point. 4,19 Alternatively, the cutoff can be due to the occurrence of a double root of the Sagdeev potential (signifying a double layer). 35 Finally, we need to be aware of the limitations imposed by the model, which themselves may give rise to an upper cut-off in speed, as we have noted above (1 < M < ffiffiffi r p ).…”

“…29,30 In order to bring out the differences between keeping and omitting the hot electron inertial effects, while treating the species as behaving isothermally, we proceed to the Bernoulli integrals 4,19,31 by integrating (4) and using mass conservation from (1). This yields a relation between n h and u…”

“…Later, this work was followed by Cattaert et al 17 and Verheest et al, 18 who used the McKenzie fluid dynamical approach 4,19 (a fully nonlinear technique that is analogous to the Sagdeev formalism), in investigating electron-acoustic solitons and double layers. They showed that both positive and negative potential electron-acoustic structures could exist in a plasma in which there were no drifts.…”