= Solution
Substitution of $e^{i(kx-\omega t)}$ gives the <dispersion relation> $\boxed{\omega(k)=Uk+k^3}$. The <phase velocity> is $U+k^2$, whereas the <group velocity> is $U+3k^2$. Thus nonzero-wavenumber crests move more slowly than their wave packet; the velocities coincide at zero wavenumber.
Fourier evolution gives $\phi(Vt,t)=\int A(k)e^{it\Phi(k)}\,dk$, where $\Phi(k)=(V-U)k-k^3$. For $V>U$ the two nondegenerate stationary points are $k=\pm\kappa$, $\kappa=\sqrt{(V-U)/3}$. At these points $\Phi(\pm\kappa)=\pm2\kappa^3$ and $\Phi''(\pm\kappa)=\mp6\kappa$. Expanding the phase quadratically near each point and evaluating the oscillatory Gaussian integral gives the <stationary phase method>
$$
\boxed{\phi(Vt,t)\sim\sqrt{\frac{\pi}{3\kappa t}}\left[
A(\kappa)e^{i(2\kappa^3t-\pi/4)}+A(-\kappa)e^{-i(2\kappa^3t-\pi/4)}\right].}
$$
For a real initial field, $A(-\kappa)=\overline{A(\kappa)}$, so the two terms are conjugate. The approximation assumes a sufficiently smooth decaying amplitude; if a leading coefficient vanishes, the next stationary-phase terms determine that contribution.
For $V<U$, $\Phi'(k)=V-U-3k^2<0$ everywhere. There are no real stationary points, so no leading propagating packet reaches such a ray. For a Schwartz amplitude, repeated integration by parts with $e^{it\Phi}=(it\Phi')^{-1}\partial_ke^{it\Phi}$ makes the contribution $O(t^{-N})$ for every fixed $N$. With weaker data the decay rate depends on their regularity. This agrees with the minimum <group velocity> being $U$.
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