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Rigorous merging of two-stream and Buneman instabilities

Two-stream and Buneman instabilities are among the most well-known streaming plasma instabilities. In general, they occur within distinct ranges of wave vectors and can be treated separately in the linear regime. For symmetric counter-streams however, these modes overlap and even merge exactly for some wavelengths. The corresponding range can be expressed using Cardano's method for the resolution of the cubic equation.