Consider the $\mathrm{KdV}$ equation for the function $u(x, t)$

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

(a) Write equation (1) in the Hamiltonian form

$$u_{t}=\frac{\partial}{\partial x} \frac{\delta H[u]}{\delta u}$$

where the functional $H[u]$ should be given. Use equation (1), together with the boundary conditions $u \rightarrow 0$ and $u_{x} \rightarrow 0$ as $|x| \rightarrow \infty$, to show that $\int_{\mathbb{R}} u^{2} d x$ is independent of $t$.

(b) Use the Gelfand-Levitan-Marchenko equation

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

to find the one soliton solution of the KdV equation, i.e.

$$u(x, t)=-\frac{4 \beta \chi \exp (-2 \chi x)}{\left[1+\frac{\beta}{2 \chi} \exp (-2 \chi x)\right]^{2}}$$

[Hint. Consider $F(x)=\beta \exp (-\chi x)$, with $\beta=\beta_{0} \exp \left(8 \chi^{3} t\right)$, where $\beta_{0}, \chi$ are constants, and $t$ should be regarded as a parameter in equation (2). You may use any facts about the Inverse Scattering Transform without proof.]