H^+(\mathbb{R}),$$ where $\mathcal{L}$ is the nonlocal selfadjoint operator . Equation determines the (trivial, as we will recall) time dependence of the scattering data.

Scattering data
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The theory of the inverse-scattering transform solution of the Cauchy problem for the BO equation was first developed by Fokas and Ablowitz [@Fokas; @1983]. Certain analytical details of the theory were clarified by Coifman and Wickerhauser [@Coifman; @1990], and more recently Kaup and Matsuno [@Kaup; @1998] found conditions on the scattering data consistent with real-valued solutions of . As an operator on $\mathbb{H}^+(\mathbb{R})$, the essential spectrum of $\mathcal{L}$ is the positive real $\lambda$-axis (for suitable ${u_\epsilon}$, $\mathcal{L}$ is a relatively compact perturbation of the “free” operator corresponding to ${u_\epsilon}\equiv 0$). For each fixed $t$ and each real $\lambda >0$, there exists a unique solution $w^+=M$ of with the property that (remarkably, despite the nonlocal nature of the problem) it is determined by its asymptotic behavior as $x\to -\infty$ on the real line: $M(x,t;\lambda)=1+o(1)$ as $x\to -\infty$. As the problem is nonlocal, $M$ cannot be characterized by a Volterra-type integral equation, but Fokas and Ablowitz [@Fokas; @1983] gave a Fredholm-type equation whose unique solution is $M$. The *reflection coefficient* for the problem is then defined 