Describe the inverse scattering transform for the $\mathrm{KdV}$ equation, paying particular attention to the Lax representation and the evolution of the scattering data.

[Hint: you may find it helpful to consider the operator

$$\left.A=4 \frac{d^{3}}{d x^{3}}-3\left(u \frac{d}{d x}+\frac{d}{d x} u\right) \cdot\right]$$