Let $\psi(k ; x, t)$ satisfy the linear integral equation

$$\psi(k ; x, t)+i e^{i\left(k x+k^{3} t\right)} \int_{L} \frac{\psi(l ; x, t)}{l+k} d \lambda(l)=e^{i\left(k x+k^{3} t\right)}$$

where the measure $d \lambda(k)$ and the contour $L$ are such that $\psi(k ; x, t)$ exists and is unique.

Let $q(x, t)$ be defined in terms of $\psi(k ; x, t)$ by

$$q(x, t)=-\frac{\partial}{\partial x} \int_{L} \psi(k ; x, t) d \lambda(k)$$

(a) Show that

$$(M \psi)+i e^{i\left(k x+k^{3} t\right)} \int_{L} \frac{(M \psi)(l ; x, t)}{l+k} d \lambda(l)=0$$

where

$$M \psi \equiv \frac{\partial^{2} \psi}{\partial x^{2}}-i k \frac{\partial \psi}{\partial x}+q \psi$$

(b) Show that

$(N \psi)+i e^{i\left(k x+k^{3} t\right)} \int_{L} \frac{(N \psi)(l ; x, t)}{l+k} d \lambda(l)=3 k e^{i\left(k x+k^{3} t\right)} \int_{L} \frac{(M \psi)(l ; x, t)}{l+k} d \lambda(l)$,

where

$$N \psi \equiv \frac{\partial \psi}{\partial t}+\frac{\partial^{3} \psi}{\partial x^{3}}+3 q \frac{\partial \psi}{\partial x}$$

(c) By recalling that the $\mathrm{KdV}$ equation

$$\frac{\partial q}{\partial t}+\frac{\partial^{3} q}{\partial x^{3}}+6 q \frac{\partial q}{\partial x}=0$$

admits the Lax pair

$$M \psi=0, \quad N \psi=0,$$

write down an expression for $d \lambda(l)$ which gives rise to the one-soliton solution of the $\mathrm{KdV}$ equation. Write down an expression for $\psi(k ; x, t)$ and for $q(x, t)$.