Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 3 / 31E / c

A simple zero at $ip$ supplies a discrete scattering mode and hence an order-one, right-moving soliton component of speed $4p^2$. Its position depends on the associated norming constant and, for general scattering data, the appropriate scattering phase shifts. A simple zero alone does not specify its position, nor exclude additional zeros and therefore additional solitons.

To locate the continuous radiation, set $\xi =x/t$. The real-axis oscillatory phase is $t(2k\xi +8k^3)$, whose stationary points satisfy $2\xi +24k^2=0$. For $\xi <0$ there are two real stationary points $k=\pm \sqrt{-\xi /12}$, giving dispersive oscillatory radiation, ordinarily of order $t^{-1/2}$ for nondegenerate data. For $\xi >0$ there is no real stationary point, so radiation is smaller and a soliton dominates near its moving centre. At $\xi =0$ the stationary points merge and the natural transition scale is $x=O(t^{1/3})$, with Airy-type radiation of order $t^{-1/3}$. These statements assume the regular decaying scattering data implicit in the reconstruction problem; vanishing reflection can remove the radiation entirely. The discrete mode travels right, while the leading continuous radiation occupies left-going rays.