= Solution
When $b=0$, put $D=e^{-2px+8p^3t-2x_0}>0$. Since $c=2ip e^{-2x_0}$, evaluation at $ip$ gives $N(ip)=1-DN(ip)$, hence
$$
N(ip)=\frac1{1+D},\qquad N(k)=1-\frac{2ipD}{(1+D)(k+ip)}.
$$
Differentiate the coefficient of $1/k$ with respect to $x$, using $D_x=-2pD$, and apply the reconstruction formula. The result is
$$
\boxed{q(x,t)=\frac{8p^2D}{(1+D)^2}=2p^2\operatorname{sech}^2(px-4p^3t+x_0).}
$$
The positive sign follows from the source's $-2i$ reconstruction convention. The <soliton> has amplitude $2p^2$, speed $4p^2$, and centre $x=4p^2t-x_0/p$.
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