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The following equation is shown to govern the finite-amplitude ion-wave propagating through a macroscopic plasma with an ion-beam in the neighborhood of a marginally stable state: \({∂²ψ}{∂τ²}+a{∂²ψ}{∂ξ²}+{∂⁴ψ}{∂ξ⁴}+{∂²ψ²}{∂ξ²}{=}0\), where a denotes the deviation of the plasma system from the marginal state. It is found that wave-trains or solitary waves exist. The temporal evolution of initial disturbances is investigated numerically.
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Kako et al. (1972) studied this question.
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