= Solution
For $V<U$,
$$
\psi'(k)=V-U-k^8<0
$$
for every real $k$, so there is no real stationary point. The ray $x=Vt$ travels more slowly than the minimum group velocity $U$ and receives no leading stationary-phase wave packet. Under the usual smoothness and decay assumptions on $A$, the <nonstationary oscillatory integral> can be integrated by parts repeatedly, so
$$
\boxed{\phi(Vt,t)=O(t^{-N})}
$$
for every fixed $N$ for which the needed derivatives are integrable. In particular it is asymptotically much smaller than the $t^{-1/2}$ signal when $V>U$.
Back to article page
