Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 3 / 31E (Integrable Systems)

Suppress $(x,t)$, put $R(k)=b(k)/a(k)$ and $C=c{e}^{-2px+8p^{3}t}N(ip)$. Removing the pole gives the additive jumpThe upper analytic function is $\mathcal{M}-1$ and the lower analytic function is $N(-k)-1$. The Cauchy integral formula and the Sokhotski–Plemelj formula solve this jump as the upper and lower boundary values of $(2\pi i{)}^{-1}{\int }_{\mathbb{R}}G(s)/(s-z)ds$. At the lower point $z=-k$, with $\Im k>0$, this gives $N(k)-1$.

The rational part can be evaluated by closing the contour upward: ${\int }_{\mathbb{R}}[(s-ip)(s+k){]}^{-1}ds=2\pi i/(k+ip)$. Thus the required linear integral equation isIt is supplemented by its value at $k=ip$, and follows under the usual decay or principal-value hypotheses of the stated Riemann-Hilbert problem.

When $b=0$, put $D=e^{-2px+8p^{3}t-2x_0}>0$. Since $c=2ip{e}^{-2x_0}$, evaluation at $ip$ gives $N(ip)=1-DN(ip)$, henceDifferentiate the coefficient of $1/k$ with respect to $x$, using $D_x=-2pD$, and apply the reconstruction formula. The result isThe positive sign follows from the source's $-2i$ reconstruction convention. The soliton has amplitude $2p^2$, speed $4p^2$, and centre $x=4p^{2}t-x_0/p$.

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.