Let $u=u(x, t)$ be a smooth solution to the $\mathrm{KdV}$ equation

$$u_{t}+u_{x x x}-6 u u_{x}=0$$

which decays rapidly as $|x| \rightarrow \infty$ and let $L=-\partial_{x}^{2}+u$ be the associated Schrödinger operator. You may assume $L$ and $A=4 \partial_{x}^{3}-3\left(u \partial_{x}+\partial_{x} u\right)$ constitute a Lax pair for KdV.

Consider a solution to $L \varphi=k^{2} \varphi$ which has the asymptotic form

$$\varphi(x, k, t)= \begin{cases}e^{-\mathrm{i} k x}, & \text { as } x \rightarrow-\infty \\ a(k, t) e^{-\mathrm{i} k x}+b(k, t) e^{\mathrm{i} k x}, & \text { as } x \rightarrow+\infty\end{cases}$$

Find evolution equations for $a$ and $b$. Deduce that $a(k, t)$ is $t$-independent.

By writing $\varphi$ in the form

$$\varphi(x, k, t)=\exp \left[-\mathrm{i} k x+\int_{-\infty}^{x} S(y, k, t) \mathrm{d} y\right], \quad S(x, k, t)=\sum_{n=1}^{\infty} \frac{S_{n}(x, t)}{(2 \mathrm{i} k)^{n}}$$

show that

$$a(k, t)=\exp \left[\int_{-\infty}^{\infty} S(x, k, t) \mathrm{d} x\right]$$

Deduce that $\left\{\int_{-\infty}^{\infty} S_{n}(x, t) \mathrm{d} x\right\}_{n=1}^{\infty}$ are first integrals of KdV.

By writing a differential equation for $S=X+\mathrm{i} Y$ (with $X, Y$ real), show that these first integrals are trivial when $n$ is even.