Let $u=u(x)$ be a smooth function that decays rapidly as $|x| \rightarrow \infty$ and let $L=-\partial_{x}^{2}+u(x)$ denote the associated Schrödinger operator. Explain very briefly each of the terms appearing in the scattering data

$$S=\left\{\left\{\chi_{n}, c_{n}\right\}_{n=1}^{N}, R(k)\right\},$$

associated with the operator $L$. What does it mean to say $u(x)$ is reflectionless?

Given $S$, define the function

$$F(x)=\sum_{n=1}^{N} c_{n}^{2} e^{-\chi_{n} x}+\frac{1}{2 \pi} \int_{-\infty}^{\infty} e^{\mathrm{i} k x} R(k) \mathrm{d} k$$

If $K=K(x, y)$ is the unique solution to the GLM equation

$$K(x, y)+F(x+y)+\int_{x}^{\infty} K(x, z) F(z+y) \mathrm{d} z=0$$

what is the relationship between $u(x)$ and $K(x, x)$ ?

Now suppose that $u=u(x, t)$ is time dependent and that it solves the KdV equation $u_{t}+u_{x x x}-6 u u_{x}=0$. Show that $L=-\partial_{x}^{2}+u(x, t)$ obeys the Lax equation

$$L_{t}=[L, A], \quad \text { where } A=4 \partial_{x}^{3}-3\left(u \partial_{x}+\partial_{x} u\right) .$$

Show that the discrete eigenvalues of $L$ are time independent.

In what follows you may assume the time-dependent scattering data take the form

$$S(t)=\left\{\left\{\chi_{n}, c_{n} e^{4 \chi_{n}^{3} t}\right\}_{n=1}^{N}, R(k, 0) e^{8 \mathrm{i} k^{3} t}\right\} .$$

Show that if $u(x, 0)$ is reflectionless, then the solution to the KdV equation takes the form

$$u(x, t)=-2 \frac{\partial^{2}}{\partial x^{2}} \log [\operatorname{det} A(x, t)]$$

where $A$ is an $N \times N$ matrix which you should determine.

Assume further that $R(k, 0)=k^{2} f(k)$, where $f$ is smooth and decays rapidly at infinity. Show that, for any fixed $x$,

$$\int_{-\infty}^{\infty} e^{\mathrm{i} k x} R(k, 0) e^{8 \mathrm{i} k^{3} t} \mathrm{~d} k=O\left(t^{-1}\right) \quad \text { as } t \rightarrow \infty$$

Comment briefly on the significance of this result.

[You may assume $\frac{1}{\operatorname{det} A} \frac{\mathrm{d}}{\mathrm{d} x}(\operatorname{det} A)=\operatorname{tr}\left(A^{-1} \frac{\mathrm{d} A}{\mathrm{~d} x}\right)$ for a non-singular matrix $A(x)$.]