Consider the equation

$$\frac{\partial \phi}{\partial t}+\frac{\partial \phi}{\partial x}-\frac{\partial^{3} \phi}{\partial x^{3}}=0$$

Find the dispersion relation for waves of frequency $\omega$ and wavenumber $k$. Do the wave crests move faster or slower than a packet of waves?

Write down the solution with initial value

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

where $A(k)$ is real and $A(-k)=A(k)$.

Use the method of stationary phase to obtain an approximation to $\phi(x, t)$ for large $t$, with $x / t$ having the constant value $V$. Explain, using the notion of group velocity, the constraint that must be placed on $V$.