Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 32C (Integrable Systems)

Normalize the solution by its incoming amplitude at $+\infty$. The transmission coefficient and reflection coefficient areFor two solutions of the same equation in Sturm-Liouville form, the modified Wronskianis independent of $x$. Take ${\psi }_1={\varphi }_k$ and ${\psi }_2=\overline{{\varphi }_k}$. At $-\infty$ this gives $W=-2ik$, while at $+\infty$ it givesTherefore $|a{|}^2-|b{|}^2=1$, and division by $|a{|}^2$ yields conservation of scattering flux:

Suppose a normalizable solution satisfies $L\psi =-{\kappa }^2\psi$. Multiplication by $\overline{\psi }$, integration over $\mathbb{R}$, and integration by parts giveThe boundary term vanishes for a normalizable eigenfunction. The left side is negative unless $\psi =0$, so no nontrivial normalizable solution exists.

Differentiatingwith respect to time givesIf $p_t=p^3{p}_{xxx}$, thenThus $L_t=[L,A]$ is an Isospectral Lax equation, and $(L,A)$ is a Lax pair for the Harry Dym equation.

As $|x|\to \infty$, the operator $A$ tends to $-4{\partial }_x^3$. Evolve the normalized scattering eigenfunction byThe added scalar term keeps the coefficient of $e^{-ikx}$ at $-\infty$ equal to one. At $+\infty$,ConsequentlyIt follows that