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On the kinetic equation for resonant three-wave coupling†

Published online by Cambridge University Press:  13 March 2009

Ronald C. Davidson  and
Allan N. Kaufman
Ronald C. Davidson
Affiliation:
University of California, Berkeley, California
Allan N. Kaufman
Affiliation:
University of California, Berkeley, California
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Abstract

The kinetic equation for resonant three-wave coupling is solved exactly in the one-dimensional case assuming that only a single triplet of modes in interacting. No assumptions are made as to the froms of the interaction kernel or of the oscillation frequencies as functions of wave-number. Both positive- and negative- energy waves are allowed to be present simultaenously. In explosively unstable situations the time, t∞, for divergent behaviour to occur is calculated in terms of the interaction keral and intial conditions. In situations where the system is non-linearly stable, the density of quasi-particles is found to tend time-asymptotically to a steady state which depends in detail on the initial preparation of the system.

For the case of two or three dimensions, the asymptotic state is obtained when the system is stable, the k-space is finite, and only one triplet of modes interact. If these modes all have the same sign of energy, the Rayleigh—Jeans distribution is valid; but if one of the modes has the oppsoite sign of energy, an additional conservation law leads to a modification of the Rayleigh-Jeans distribution.

Type
Research Article
Information
Journal of Plasma Physics , Volume 3 , Issue 1 , February 1969 , pp. 97 - 105
Copyright
Copyright © Cambridge University Press 1969

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Footnotes

‡Present address: University of Maryland, College Park, Maryland. 7/68.
†

This work was supported in part by the National Aeronautics and Space Adrninistration, Grant NGR-05-003-220, and in part by the United States Atomic Energy Commission

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