= Solution
Retain $K=k^2\lambda_{\mathrm{De}}^2$ while making the same narrow-<velocity plateau> expansion as in <2/d>. With $u=v_1+\delta a$, the factor that varies rapidly across the plateau is $(v_2-u)/(u-v_1)=(1-a)/a$. The other factors vary slowly and may be evaluated at $u=v_1$, giving
$$
\frac{v_1^2}{v_0^2-v_1^2}\frac{1-a}{a}=e^K.
$$
Therefore
$$
\boxed{\omega_{\mathrm a}\approx k\left[v_1+\frac{v_2-v_1}{1+e^{k^2\lambda_{\mathrm{De}}^2}(v_0^2-v_1^2)/v_1^2}\right].}
$$
At $K\ll1$, the denominator tends to $v_0^2/v_1^2$, recovering <2/d>. At $K\gg1$,
$$
u_{\mathrm a}-v_1\approx\delta\frac{v_1^2}{v_0^2-v_1^2}e^{-K},
$$
recovering the exponentially close approach to the plateau edge in <2/c>. The <phase velocity> remains strictly inside the <velocity plateau> for every finite $k$.
As an asymptotic expansion, the formula takes $\delta\to0$ at fixed $0<v_1/v_0<1$. If the plateau also approaches the outer edge, so that $v_0-v_1$ is comparable to $\delta$, the factors treated as slowly varying need not be slowly varying. In that additional limiting regime, use the exact <dispersion relation> in <2/b> instead.
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