Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 33E (Integrable Systems)

Since $L^T=L$ and $B^T=-B$,so the commutator is symmetric. If $Lv=\lambda v$ with $\Vert v\Vert =1$, symmetry permits a differentiable orthonormal eigenbasis locally, andThus a Lax pair evolution is isospectral.

For the harmonic oscillator,With the stated $L$, takeDirect calculation givesThe eigenvalues of $L$ areTheir constancy proves conservation of the oscillator energy $H={\lambda }_{+}^2/2$.

Differentiating the first auxiliary equation in $t$, the second in $x$, and equating ${\psi }_{xt}$ and ${\psi }_{tx}$ leaves preciselyso the Airy equation is their compatibility condition.

Multiplication of ${\psi }_x-ik\psi =q$ by $e^{-ikx}$ givesHenceFor $\mathrm{Im}k\ge 0$, the kernel $e^{ik(x-y)}$ is bounded when $y\le x$; dominated differentiation makes ${\psi }_{+}$ analytic for $\mathrm{Im}k>0$. Rapid decrease gives

For real $k$, multiply the time equation by $e^{-ikx}$ and let $x\to +\infty$. The rapidly decreasing terms on its right vanish, leavingThereforeand the Fourier inversion theorem gives