of periodic traveling wave solutions of the BO equation , along with quasiperiodic generalizations. Remarkably, unlike the corresponding exact solutions of the KdV equation which are highly transcendental objects constructed from Riemann theta functions of hyperelliptic curves of arbitrary genus, the periodic and quasiperiodic solutions of the BO equation turn out to be simple rational functions of $P$ exponential phases $e^{i(k_jx-\omega_jt)/\epsilon}$. In the same paper, Dobrokhotov and Krichever also carried out for the BO equation the analogue of the calculation of Flaschka, Forest, and McLaughlin [@FlaschkaFM80], deriving by multiphase averaging a system of equations governing the modulational variables for a slowly-varying train of $P$-phase waves. Here we arrive at a second remarkable fact: not only can the modulation equations be written in Riemann invariant form, they are completely diagonal: $$\frac{\partial u^i}{\partial t} + 2u^i\frac{\partial u^i}{\partial x}=0,
\quad i=1,\dots,2P+1$$ (the case of $P=1$ corresponds to simple traveling waves). This again should be contrasted with the situation for the KdV equation in which the characteristic velocities not only provide coupling among the fields but also are transcendental functions of the fields written in terms of ratios of complete hyperelliptic integrals.

The analogue for the BO equation of the matching procedure developed by 