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The kinetic theory of ion-acoustic waves in multi-ion-species plasmas is discussed. Particular application is made to hydrocarbon (CH) plasmas, which are widely used in laser–plasma experiments. The mode frequencies and Landau damping of the two, dominant, ion-acoustic modes in CH plasmas are calculated by numerical solution of the kinetic dispersion relation. In addition, some useful results are obtained analytically from expansions of the kinetic dispersion relation and from fluid models. However, these results disagree with the numerical results in domains of particular practical interest. When ion temperatures exceed two-tenths of the electron temperature, the least damped mode is the one with the smaller phase velocity, and this mode is then found to dominate the ponderomotive response of the CH plasma.

When an instability mode which is driven unstable can have its growth reversed because it is depleted by driving (pumping) another similar mode via a three-mode interaction, and when that process can be repeated over several pump modes, one can speak of an instability cascade. If the modes that are not acting as pumps are heavily damped (as in nonlinear Landau damping for plasma waves), the relative phases have no interesting behavior and the equations for the* mode actions are a particular case of the general Lotka-Volterra interaction system, that of a nearest-neighbor cascade. 1 (This can be thought of as a system of predation where each population species preys only on the population next nearest the ultimate source and is being preyed on only by the population next further from the source.)The general Lotka-Volterra equation system 2 is as follows, for the variables n i9 their natural linear growth rates k i9 and interaction parameters a u and 0,:Here n i (which must be positive to be acceptable) can represent the ith concentration of, for instance, species, chemical component, or wave action. The a u is antisymmetric, reflecting the zero-sum nature of the ij pair interactions, while the j3, coefficients (which are positive) are the Volterra species equivalence numbers. For chemistry the p { coefficients are such that ^i/fi i is the ratio of the number of molecules involved in the ij interaction, and for three-wave interactions all the P t coefficients are 1 (from the Manley-Rowe relations). The original two-speciesLett. 6>, 411 (1981); M. LeBerre, E. Ressayre, and A. Tallet, Phys. Rev. A'25, 1604 (1982), and references therein.version of these equations was invoked by Lotka 2 for autocatalytic reactions and by Volterra 2 to explain fluctuations in fish species in the Adriatic sea, while several authors have used them to consider plasma-wave instability competition. 1 On numerical investigation, it became evident that the time behavior could be remarkably irregular. Using Poincarg surfaces of section we found the behavior to resemble that of Hamilto-7 nian chaos, 3 and then realized that the system was indeed Hamiltonian. We recapitulate for convenience an earlier treatment 2 of an even number of equations, with a slight (but important) variation, because this leads naturally to the Hamiltonian. We definewhere y { is the formal equilibrium value of n { . The y { hence must satisfy Eq. (1) with h i set equal to zero.For an odd number of equations the determinant of the antisymmetric a u is necessarily zero, which is why we consider only even systems with IT members and invertible a { /a. Note that, unlike n/s, some y/s may be negative. (We return to this point later when discussing nearestneighbor cascades.) It has been shown 2 that there is a general Lotka-Volterra invariant, which we call H because it is in truth a Hamiltonian, given by ^EiMwi-^lnwj+yftal-yJ).(2)We have used w i here (rather than the earlier 2 w.i/Pi) because this allows the following attractive form for the equations when H is expressed Pa...

We are indebted to Dr. Luc Berge who has brought to our attention the fact that we carelessly reproduced a trivial error of Zakharov et al. [1]. The correct quantity to be varied about its minimum is, of course, H 1 lN and not 2H 1 lN (as we wrote) nor the quantity H 2 lN given previously [1].[1] V. E. Zakharov, V. V. Sobolev, and V. C. Synakh, Zh. Eksp. Teor. Fiz. 60, 136 (1971) [Sov. Phys. JETP 33, 77 (1971]. 48520031-9007͞96͞77(23)͞4852(1)$10.00