solution of the Cauchy problem for the KdV equation with smooth, $\epsilon$-independent initial data $v_0(x)$ really does generate after some fixed breaking time a train of high-frequency waves of exactly the kind originally considered without complete justification by Whitham. More recently, the powerful steepest-descent method for matrix Riemann-Hilbert problems developed by Deift and Zhou was used to analyze the zero-dispersion limit for the KdV equation [@Deift; @1997]. This technique is best viewed as a tool for converting weak asymptotics (the solution of the Lax-Levermore variational problem) into strong asymptotics (an improvement of the Venakides asymptotics in which the phase of the waveform is accurate to very high order).

The zero-dispersion limit of the Benjamin-Ono equation
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It turns out that the BO equation is also an integrable equation, in the sense that it has a representation as the compatibility condition of an overdetermined Lax pair of linear problems [@Bock; @1979]. In fact, both BO and KdV equations may be viewed as limiting cases (as depth of a fluid layer tends to infinity and zero, respectively) of the so-called intermediate long-wave equation [@KodamaAS82], itself an integrable system for arbitrary layer depth. However, the integrable structure of the BO equation is markedly 