Fundamentals of Plasma Physics 2025, 45 - Instability Threshold

Damping can be introduced through an electrical circuit consisting of an inductor, capacitor and resistor in series. The circuit is defined by

Solving this dispersion relation for the instability threshold's position, and frequency mismatch Δ = ω₃ - (ω₁ + ω₂) gives

resulting in a growth rate near the threshold of

The low-frequency daughter mode at 0 frequency, the ion acoustic mode ceases to be a wave, and becomes a density depletion, which can be written as a Boltzmann-like relation via integration. The high-frequency daughter wave is the qualitatively the same as the pump wave, so the modes don't need distinguishing. An undamped linear Langmuir wave in a uniform plasma has a dispersion relation ω² = ω²_pe(1+3k²λ²_De), kλ_De ≪ 1, with wave frequency close to ω_pe.

Using the analogy of a kind of Schroedinger equation and classical conservation of energy relation for a particle in a potential well V(x). For a caviton, assume a stable solution χ₀ that is bounded in both time and space. Assume a slightly different solution χ with a perturbation term, which is very small compared to to χ₀. Assume the perturbation is unstable, with space-time dependence of some periodic exponential, then the full dispersion relation for the growth rate comes out to (γ+η)² = -k⁴ + 4k²|χ₀|² - 3|χ₀|⁴ with a maximum γ identified analytically to γ_max = -η + |χ₀|². A stationary envelope soliton for the Schroedinger-type equation has a solution of form χ = g(ξ)e^iΩτ, which gives a differential equation -Ωg + g³ + g'' = 0. The solutions to the systems are as follows.

The soliton may propagate by addition of a velocity into the solution: χ = g(ξ-vt)e^{iΩτ + ih(ξ, τ)}