The function $\phi(x, t)$ satisfies the equation

$$\frac{\partial \phi}{\partial t}+U \frac{\partial \phi}{\partial x}+\frac{1}{5} \frac{\partial^{5} \phi}{\partial x^{5}}=0$$

where $U>0$ is a constant. Find the dispersion relation for waves of frequency $\omega$ and wavenumber $k$. Sketch a graph showing both the phase velocity $c(k)$ and the group velocity $c_{g}(k)$, and state whether wave crests move faster or slower than a wave packet.

Suppose that $\phi(x, 0)$ is real and given by a Fourier transform as

$$\phi(x, 0)=\int_{-\infty}^{\infty} A(k) e^{i k x} d k$$

Use the method of stationary phase to obtain an approximation for $\phi(V t, t)$ for fixed $V>U$ and large $t$. If, in addition, $\phi(x, 0)=\phi(-x, 0)$, deduce an approximation for the sequence of times at which $\phi(V t, t)=0$.

What can be said about $\phi(V t, t)$ if $V<U$ ? [Detailed calculation is not required in this case.]

[You may assume that $\int_{-\infty}^{\infty} e^{-a u^{2}} d u=\sqrt{\frac{\pi}{a}}$ for $\operatorname{Re}(a) \geqslant 0, a \neq 0 .$ ]