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

$$\frac{\partial \phi}{\partial t}+U \frac{\partial \phi}{\partial x}=\frac{\partial^{3} \phi}{\partial x^{3}},$$

where $U>0$ is a constant. Find the dispersion relation for waves of wavenumber $k$ and deduce whether wave crests move faster or slower than a wave packet.

Suppose that $\phi(x, 0)$ is 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 find $\phi(V t, t)$ as $t \rightarrow \infty$ for fixed $V>U$.

[You may use the result that $\int_{-\infty}^{\infty} e^{-a \xi^{2}} d \xi=(\pi / a)^{1 / 2}$ if $\left.\operatorname{Re}(a) \geqslant 0 .\right]$

What can be said if $V<U$ ? [Detailed calculation is not required in this case.]