is to determine the solution ${u_\epsilon}(x,t)$ of subject to a suitable initial condition ${u_\epsilon}(x,0)=u_0(x)$ given for all $x\in\mathbb{R}$.

The parameter $\epsilon>0$ is a measure of the relative strength of the dispersive and nonlinear effects in the system. In many applications one thinks of $\epsilon$ as a small parameter in part because numerical experiments show that in this situation the finite-time formation of a shock wave (gradient catastrophe) in the formal limiting equation (obtained simply by setting $\epsilon=0$ in ) is dispersively regularized by the generation of a smoothly modulated train of approximately periodic traveling waves, which correspond to so-called undular bores frequently observed in the evolution of physical internal waves. Snapshots from the solution of a Cauchy problem for illustrating the averted shock and onset of an undular bore are shown in Figure \[fig:numerics1\].

![*The evolution of a pulse under the BO equation. Top row: $\epsilon=0.04$. Bottom row: $\epsilon=0.02$. In both cases the initial condition is the same: $u_0(x)=2(1+x^2)^{-1}$.*[]{data-label="fig:numerics1"}](BOFig1.pdf)

These figures clearly show that the mathematical description of the undular bore consists of waves of amplitude independent of $\epsilon$ and wavelength approximately proportional to $\epsilon$. We refer to the asymptotic analysis of the solution of the Cauchy problem with $\epsilon$-independent 