The solution of the initial value problem of the $\mathrm{KdV}$ equation is given by

$$q(x, t)=-2 i \lim _{k \rightarrow \infty} k \frac{\partial N}{\partial x}(x, t, k),$$

where the scalar function $N(x, t, k)$ can be obtained by solving the following RiemannHilbert problem:

$$\frac{M(x, t, k)}{a(k)}=N(x, t,-k)+\frac{b(k)}{a(k)} \exp \left(2 i k x+8 i k^{3} t\right) N(x, t, k), \quad k \in \mathbb{R},$$

$M, N$ and $a$ are the boundary values of functions of $k$ that are analytic for $\operatorname{Im} k>0$ and tend to unity as $k \rightarrow \infty$. The functions $a(k)$ and $b(k)$ can be determined from the initial condition $q(x, 0)$.

Assume that $M$ can be written in the form

$$\frac{M}{a}=\mathcal{M}(x, t, k)+\frac{c \exp \left(-2 p x+8 p^{3} t\right) N(x, t, i p)}{k-i p}, \quad \operatorname{Im} k \geqslant 0,$$

where $\mathcal{M}$ as a function of $k$ is analytic for $\operatorname{Im} k>0$ and tends to unity as $k \rightarrow \infty ; c$ and $p$ are constants and $p>0$.

(a) By solving the above Riemann-Hilbert problem find a linear equation relating $N(x, t, k)$ and $N(x, t, i p)$.

(b) By solving this equation explicitly in the case that $b=0$ and letting $c=2 i p e^{-2 x_{0}}$, compute the one-soliton solution.

(c) Assume that $q(x, 0)$ is such that $a(k)$ has a simple zero at $k=i p$. Discuss the dominant form of the solution as $t \rightarrow \infty$ and $x / t=O(1)$.