Nonlinear instabilities, negative energy modes and generalized Cherry oscillators
In 1925 Cherry discussed two oscillators of positive and negative energy that are nonlinearly coupled in a special way, and presented a class of exact solutions of the nonlinear equations showing explosive instability independent of the strength of the nonlinearity and the initial amplitudes. In this paper Cherry's Hamiltonian is transformed into a form which allows a simple physical interpretation . The new Hamiltonian is generalised to three nonlinearly coupled oscillators; it corresponds to three-wave interaction in a continuum theory, like the Vlasov-Maxwell theory, if there exist linear negative energy waves. Cherry was able to present a two-parameter solution set for his example which would, however, allow a four-parameter solution set. As a first result, an analogous three-parameter solution set for the resonant three-oscillator case is obtained which would allow a six-parameter solution set. Nonlinear instability is therefore proven so far only for a very small part of the phase space of the oscillators. This paper gives in addition the complete solution for the three-oscillator case and shows that, except for a singular case, all initial conditions, especially those with arbitrarily small amplitudes, lead to explosive behaviour. This is true of the resonant case. The non-resonant oscillators can sometimes also become explosively unstable, but the initial amplitudes must not be infinitesimally small. A few examples are presented for illustration.
Bibliographic Reference: Article: Zeitschrift für Naturforschung, Vol. 45a (1990) pp. 839-846
Record Number: 199011461 / Last updated on: 1994-12-02
Original language: en
Available languages: en