Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 34D (Integrable Systems)

The Lax equation $L_t=[A,L]$ withreduces, after expanding the commutator, to the KdV equationThus the evolution is isospectral: the eigenvalues of $L$ are constant. The fourth derivative shown in the converted question is a transcription error.

The scattering data for the Schrodinger operator consist of the reflection coefficient $R(k,t)$ on the continuous spectrum, the negative discrete eigenvalues $-{\kappa }_n^2$, and their norming constants $c_n$. Isospectrality givesComparing the asymptotic plane waves in ${\psi }_t=A\psi$ gives

Each discrete term satisfiesand the same identity follows for the Fourier integral from $R_t=8i{k}^{3}R$. HenceApply ${\partial }_x^2-{\partial }_y^2-u(x,t)$ to the Gelfand-Levitan-Marchenko equation. Differentiating under the integral and integrating the $z$ derivatives by parts, the boundary terms combine to $u=-2{\partial }_{x}K(x,x,t)$. The terms containing $F$ cancel, leavingUniqueness of the Marchenko equation then also gives $G=0$.

Useand differentiate the spatial equationto eliminate the mixed derivative in the time equation. After collecting the terms involving $u$ and $u_x$, one obtainsSet $y=x$, use $u=-2{\partial }_{x}K(x,x,t)$ with the derivative along the diagonal, and differentiate once more in $x$. The resulting compatibility condition is