= Solution
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. \b[The discrete mode travels right, while the leading continuous radiation occupies left-going rays.]
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