= Solution
For the two outer branches, $|u|\gg v_0$ at small $k$. An <integration by parts> and a large-$u$ expansion give
$$
I(u)=\int\frac{f_0(v)}{(v-u)^2}\,dv=\frac{n_*}{u^2}+O(u^{-3}),\qquad n_*=\int f_0(v)\,dv.
$$
There is no boundary contribution because the <phase-space distribution function> vanishes at both endpoints. Direct area integration gives
$$
n_*=A_0\left[v_0+\delta\left(1-\frac{v_1}{v_0}\right)\right],\qquad \frac{n_*}{n_0}=1+\frac{\delta(v_0-v_1)}{v_0^2}.
$$
Thus the exact small-$k$ cutoff, retaining the paper's definition $n_0=v_0A_0$, is $\omega\to\pm\omega_{\mathrm{pe}}\sqrt{n_*/n_0}$. To leading order in the narrow-<velocity plateau> approximation this is
$$
\boxed{\omega\approx\pm\omega_{\mathrm{pe}}.}
$$
These are <Langmuir waves>. The difference between $n_*$ and the reference $n_0$ is only first order in $\delta/v_0$, so it does not alter the requested leading-order result.
For the finite-<phase velocity> branch put $u=v_1+\delta a$, with $0<a<1$. To leading order in $\delta/v_0$ and $\delta/v_1$, the real <dispersion relation> becomes
$$
\ln\left[\frac{v_1^2}{v_0^2-v_1^2}\frac{1-a}{a}\right]=K.
$$
At long wavelength set $K=0$ to leading order. Then $a=v_1^2/v_0^2$, and
$$
\boxed{\omega_{\mathrm a}\approx k\left[v_1+\delta\frac{v_1^2}{v_0^2}\right].}
$$
For comparison, the exact $k\to0$ <phase velocity> from the <cubic equation> in <2/b> is $B/A$; expanding it gives precisely $v_1+\delta v_1^2/v_0^2+O(\delta^2/v_0)$ at fixed $v_1/v_0$.
A single <Maxwell-Boltzmann velocity distribution> has a nonzero slope at every finite nonzero $u$ and has no <velocity plateau>. The would-be <electron acoustic wave> is therefore subject to strong <Landau damping>, so there is no corresponding completely undamped acoustic branch in a single <Maxwellian> electron population with immobile ions. At its only stationary point $u=0$, a <Maxwellian> has $\int f_0'(v)/v\,dv<0$, making the static <dielectric function> positive rather than zero. This statement does not exclude acoustic modes of suitably distinct electron populations or an ion-supported mode when ion dynamics is retained.
Back to article page
