Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 61 / 2 / a / Solution

Write $E_k=e^{i(kx+k^{3}t)}$, $I={\int }_L\phi (l)d\lambda (l)$ and $A_k={\int }_L\phi (l)/(k+l)d\lambda (l)$. Introduce the dressing operatorThe given linear integral equation is $\mathcal{F}\phi =E$, and $q=-I_x$. The appropriate contour and measure must permit differentiation under these integrals and avoid untreated kernel singularities.

For the specified spatial operator, direct differentiation givesInside the integral, replace ${\phi }_{xx}(l)+q\phi (l)$ by $M_l\phi (l)+il{\phi }_x(l)$. It follows thatApply $M_k$ to $\phi +i{E}_k{A}_k=E_k$. The last term contributes $-E_k{I}_x=q{E}_k$, canceling the right side, soThis is precisely the homogeneous integral equation. Uniqueness of the original inhomogeneous solution implies a trivial homogeneous null space, hence $M\phi =0$. This calculation is the spatial identity in Cauchy-kernel dressing for the KdV equation.