(i) Explain how the inverse scattering method can be used to solve the initial value problem for the $\mathrm{KdV}$ equation

$$u_{t}+u_{x x x}-6 u u_{x}=0, \quad u(x, 0)=u_{0}(x)$$

including a description of the scattering data associated to the operator $L_{u}=-\partial_{x}^{2}+u(x, t)$, its time dependence, and the reconstruction of $u$ via the inverse scattering problem.

(ii) Solve the inverse scattering problem for the reflectionless case, in which the reflection coefficient $R(k)$ is identically zero and the discrete scattering data consists of a single bound state, and hence derive the 1-soliton solution of $\mathrm{KdV}$.

(iii) Consider the direct and inverse scattering problems in the case of a small potential $u(x)=\epsilon q(x)$, with $\epsilon$ arbitrarily small: $0<\epsilon \ll 1$. Show that the reflection coefficient is given by

$$R(k)=\epsilon \int_{-\infty}^{\infty} \frac{e^{-2 i k z}}{2 i k} q(z) d z+O\left(\epsilon^{2}\right)$$

and verify that the solution of the inverse scattering problem applied to this reflection coefficient does indeed lead back to the potential $u=\epsilon q$ when calculated to first order in