The ion-acoustic soliton: A gas-dynamic viewpoint

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, $M_c,$ 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, $M_{\mathrm{ep}},$ in which solitons exist, is extended beyond the classical range ${\text{1<M}}_{\mathrm{ep}}\text{<1.6.}$ 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 ${\mathrm{sech}}^2$ shaped pulses characteristic of weakly nonlinear waves and shows that solitons exist only if ${\text{1<M}}_{\mathrm{ep}}\text{<2.}$ The corresponding maximum potential, associated with the critical ion-acoustic Mach number, can be between ${\text{1.3kT}}_e$ and ${\text{10kT}}_e$ depending upon the values of the adiabatic indices of the electrons and protons and the proton Mach number.