Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 3 / 37E (Waves)

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 $\varphi (Vt,t)=\int A(k)e^{it\Phi (k)}dk$, 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 $\Phi (\pm \kappa )=\pm 2{\kappa }^3$ and ${\Phi }^{\prime \prime }(\pm \kappa )=\mp 6\kappa$. Expanding the phase quadratically near each point and evaluating the oscillatory Gaussian integral gives the stationary phase methodFor 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 $V<U$, ${\Phi }^{\prime }(k)=V-U-3k^2<0$ 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 $e^{it\Phi }=(it{\Phi }^{\prime }{)}^{-1}{\partial }_k{e}^{it\Phi }$ 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 $U$.