Theory of Stability of Large Periodic Plasma Waves

Abstract

A method is developed for examining the stability of a large‐amplitude periodic Bernstein‐Greene‐Kruskal wave, $E_0$ , in a collisionless plasma. Vlasov's equation is integrated by the method of characteristics to yield a polarization charge density response ${\rho }_1$ , linear in a small‐amplitude field $E_1$ , but nonlinear in $E_0$ . The susceptibility linking ${\rho }_1$ and $E_1$ is expressed in terms of the exact orbits of trapped and untrapped particles in the field $E_0$ , distributed in energy according to an assumed Bernstein‐Greene‐Kruskal distribution function $f_0$ . These susceptibilities couple the Fourier components of $E_1$ in the usual mode‐coupling fashion, but trapping effects are now included. For fields $E_0$ which are not too large, the mode‐coupling problem reduces to finding the zeroes of a $\text{2\hspace{0.17em}×\hspace{0.17em}2}$ or $\text{3\hspace{0.17em}×\hspace{0.17em}3}$ determinant. Trapped electron distribution functions which are localized at the bottom of the potential energy troughts of $E_0$ give the growing side‐band instability of Kruer, Dawson, and Sudan.