Nonlinear interaction between two electrostatic harmonics in a plasma

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Abstract
The problem of harmonic and subharmonic generation of electrostatic waves in a general collisionless plasma, or in some other electromagnetic medium, is treated anew, using coupled-mode theory based on two time scales. A novel feature is that one of the two interacting waves may be a negative energy wave. Furthermore, the model describing the medium need not be specified, only a general linear and nonlinear conductivity or an equivalent description is required. Just by invoking wave energy conservation, the coupled-mode equations are obtained in such a way that unequivocal conclusions can be drawn. When both waves have positive energy, they exchange part of it in a periodic fashion, provided they both have some energy initially. If initially all the energy is in the fundamental, then all of it will eventually end up (irreversibly) in thesecond harmonic. If, on the other hand, all the energy is in the harmonic initially, then no generation of the fundamental (or subharmonic) will take place. If one of the two waves is a negative-energy wave, however, an explosive instability develops, and this regardless of initial values. For comparable conditions, the instability time depends on whether the negative energy wave is in the fundamental or in the upper harmonic.