(a) Let $\mathcal{L}, \mathcal{A}$ be two families of linear operators, depending on a parameter $t$, which act on a Hilbert space $H$ with inner product $(,$, . Suppose further that for each $t, \mathcal{L}$ is self-adjoint and that $\mathcal{A}$ is anti-self-adjoint. State $L a x$ 's equation for the pair $\mathcal{L}, \mathcal{A}$, and show that if it holds then the eigenvalues of $\mathcal{L}$ are independent of $t$.

(b) For $\psi, \phi: \mathbb{R} \rightarrow \mathbb{C}$, define the inner product:

$$(\psi, \phi):=\int_{-\infty}^{\infty} \overline{\psi(x)} \phi(x) d x$$

Let $L, A$ be the operators:

$$\begin{gathered} L \psi:=i \frac{d^{3} \psi}{d x^{3}}-i\left(q \frac{d \psi}{d x}+\frac{d}{d x}(q \psi)\right)+p \psi \\ A \psi:=3 i \frac{d^{2} \psi}{d x^{2}}-4 i q \psi \end{gathered}$$

where $p=p(x, t), q=q(x, t)$ are smooth, real-valued functions. You may assume that the normalised eigenfunctions of $L$ are smooth functions of $x, t$, which decay rapidly as $|x| \rightarrow \infty$ for all $t$.

(i) Show that if $\psi, \phi$ are smooth and rapidly decaying towards infinity then:

$$(L \psi, \phi)=(\psi, L \phi), \quad(A \psi, \phi)=-(\psi, A \phi)$$

Deduce that the eigenvalues of $L$ are real.

(ii) Show that if Lax's equation holds for $L, A$, then $q$ must satisfy the Boussinesq equation:

$$q_{t t}=a q_{x x x x}+b\left(q^{2}\right)_{x x}$$

where $a, b$ are constants whose values you should determine. [You may assume without proof that the identity:

$$L A \psi=A L \psi-3 i\left(p_{x} \frac{d \psi}{d x}+\frac{d}{d x}\left(p_{x} \psi\right)\right)+\left[q_{x x x}-4\left(q^{2}\right)_{x}\right] \psi$$

holds for smooth, rapidly decaying $\psi .]$