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

For a single spectral mass, the algebraic equation givesDifferentiate the reconstruction, writing $r=cE/(2\kappa )$:With $x_0={\kappa }^{-1}\log [c/(2\kappa )]$, the concise one-soliton answer isIts speed is ${\kappa }^2$ and its maximum is ${\kappa }^2/2$. Equivalently, setting $\eta =\kappa /2$ gives $q=2{\eta }^2{\mathrm{sech}}^2[\eta (x-4{\eta }^{2}t-x_0)]$. To check the sign and speed independently, this profile obeys $q_{xx}={\kappa }^{2}q-3q^2$ and $q_t=-{\kappa }^2{q}_x$; differentiation gives the stated KdV equation exactly.