for positive real $\lambda$ by the formula [@Fokas; @1983; @Kaup; @1998] $$\beta(\lambda,t):= \frac{i}{\epsilon}
\int_\mathbb{R}{u_\epsilon}(x,t)M(x,t;\lambda)e^{-i\lambda x/\epsilon}\,dx,\quad
\lambda>0.$$ For each fixed $x$ and $t$ the function $M(x,t;\lambda)$ can be shown to be the boundary value taken on the positive half-line $\lambda\in
\mathbb{R}_+$ from the upper half $\lambda$-plane of a function $W(x,t;\lambda)$ that is meromorphic in the complex $\lambda$-plane with $\mathbb{R}_+$ (a branch cut) deleted. Fokas and Ablowitz refer to the boundary value taken by $W(x,t;\lambda)$ on the positive half-line from the *lower* half-plane as $\overline{N}(x,t;\lambda)$. The poles of $W(x,t;\lambda)$ are all on the negative real $\lambda$-axis (by self-adjointness of $\mathcal{L}$) and correspond to the point spectrum of $\mathcal{L}$. It turns out that one of the consequences of the Lax pair equation is that the point spectrum is independent of time $t$. In [@Fokas; @1983] it is shown that in the generic case when $\lambda_n<0$ is a simple pole of $W$, the first two terms in the Laurent expansion of $W$ at $\lambda=\lambda_n$ are both proportional to the same function $\Phi_n(x,t)\in \mathbb{H}^+(\mathbb{R}_x)$, which is an eigenfunction of $\mathcal{L}$ with eigenvalue $\lambda=\lambda_n$. The ratio of these two terms is in fact linear in $x$: $$W(x,t;\lambda)=-i\epsilon\frac{\Phi_n(x,t)}{\lambda-\lambda_n} + 
(x+\alpha_n(t))\Phi_n(x,t)
+ O(\lambda-\lambda_n),\quad \lambda\to \lambda_n.
\label{eq:Laurent}$$ Kaup and 