Analytic BGK modes and their modulational instability

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Abstract
Physically acceptable solutions of the time-independent Vlasov–Poisson system are found by applying a method which is analogous to the soliton method in Korteweg–de Vries theory. Cnoidal wave solutions for small-amplitude Langmuir and ion acoustic waves are derived, in which the nonlinearity is determined by trapped electrons. The class of weakly distorted waves is shown to be modulational unstable.