Matsuno [@Kaup; @1998] showed that for real $u$, the complex-valued phase shift $\alpha_n(t)$ may be written in the form $$\alpha_n(t) = \gamma_n(t) -\frac{i}{2\lambda_n},\quad \gamma_n(t)\in\mathbb{R}.$$ The set of *scattering data* corresponding to the potential ${u_\epsilon}(\cdot,t)$ then consists of:

-   The reflection coefficient $\beta(\lambda,t)$ for $\lambda>0$. We write $\beta(\lambda)$ for $\beta(\lambda,0)$.

-   The negative discrete eigenvalues $\{\lambda_n\}_{n=1}^N$, $\lambda_1<\lambda_2<\cdots <\lambda_N<0$.

-   The real *phase constants* $\{\gamma_n(t)\}_{n=1}^N$. We write $\gamma_n$ for $\gamma_n(0)$.

Time dependence of the scattering data and the inverse scattering transform
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As time varies, one may expect the scattering data to vary, but the time dependence as implied by turns out to be very simple. As pointed out above, the discrete eigenvalues $\{\lambda_n\}_{n=1}^N$ are constants of the motion, and Fokas and Ablowitz [@Fokas; @1983] showed that $$\beta(\lambda,t) = \beta(\lambda)e^{i\lambda^2t/\epsilon},\quad \lambda>0,
\label{eq:timevariationbeta}$$ and $$\gamma_n(t)=\gamma_n + 2\lambda_nt,\quad n=1,\dots,N.
\label{eq:timevariationgamman}$$

The inverse-scattering procedure for solving the Cauchy problem for the BO equation with suitable real initial data $u_0(x)$ is then to calculate the scattering data at time $t=0$ from $u_0$, evolve the scattering data forward in time $t$ by the explicit formulae and , and then solve the inverse problem of constructing ${u_\epsilon}(\cdot,t)$ from the scattering data at time $t$. Generally, this requires solving 