BO equation , consists of the two equations $$\label{Laxpair1} i\epsilon \frac{\partial w^+}{\partial x}+\lambda (w^{+}-w^{-})=-{u_\epsilon}w^{+}$$ $$\label{Laxpair2} i\epsilon \frac{\partial w^\pm}{\partial t}-
2i\lambda \epsilon \frac{\partial w^\pm}{\partial x}+\epsilon^2
\frac{\partial^2 w^\pm}{\partial x^2}-2i\mathcal{C}_{\pm}
\left[\epsilon \frac{\partial {u_\epsilon}}{\partial x}\right]w^{\pm}=0,$$ where $\lambda\in\mathbb{C}$ is a spectral parameter, ${u_\epsilon}={u_\epsilon}(x,t)$ is a solution of , and $w^\pm=w^\pm(x,t;\lambda)$ are functions that are required to be, for each fixed $t$ and $\lambda$, the boundary values on the real $x$-axis of functions analytic in the upper ($+$) and lower ($-$) half complex $x$-plane. Also, $\pm \mathcal{C}_\pm =
\tfrac{1}{2}\mathbb{I}\mp\tfrac{1}{2}i\mathcal{H}$ are the orthogonal and complementary ($\mathcal{C}_+-\mathcal{C}_-=\mathbb{I}$, the Plemelj formula) projections from $L^2(\mathbb{R})$ onto its upper and lower Hardy subspaces $\mathbb{H}^\pm(\mathbb{R})$. From the point of view of the inverse-scattering transform, *i.e.* using the Lax pair as a tool to solve the Cauchy problem, equation may be considered for fixed time $t$ and defines the scattering data associated with ${u_\epsilon}(x,t)$ at time $t$. The function $w^-$ may be viewed as a kind of Lagrange multiplier present to satisfy the constraint that $w^+$ be an “upper” function. In fact, if $w^\pm\in \mathbb{H}^\pm$ then by applying $\mathcal{C}_+$ to and using the projective identities $\mathcal{C}_+[w^+]=w^+$ and $\mathcal{C}_+[w^-]=0$, can be written in the form of an eigenvalue problem $$\mathcal{L}w^+ = \lambda w^+,\quad w^+\in 