= Solution
For $0<V<\beta/(8\ell^2)$ there are two positive solutions $k_1\in(\ell,\sqrt3\ell)$ and $k_2>\sqrt3\ell$, together with their negatives. The second <derivative> is
$$
\omega''(k)=\frac{2\beta k(3\ell^2-k^2)}{(k^2+\ell^2)^3}.
$$
Each negative <stationary point> has the conjugate contribution of its positive partner, because $A$ is real even and $\Phi_V$ is odd. Therefore
$$
\boxed{\varphi(Vt,t)\sim2\sqrt{\frac{2\pi}{t}}\left[
\frac{A(k_1)}{\sqrt{|\omega''(k_1)|}}\cos\left(t\Phi_V(k_1)-\frac\pi4\right)+
\frac{A(k_2)}{\sqrt{|\omega''(k_2)|}}\cos\left(t\Phi_V(k_2)+\frac\pi4\right)\right].}
$$
If a leading amplitude vanishes, this gives a zero coefficient and a higher-order term supplies that branch's first nonzero contribution.
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