which the three equations are only coupled through the characteristic velocities: $$\frac{\partial v^i}{\partial t} + c^i(v^1,v^2,v^3)
\frac{\partial v^i}{\partial x}=0,\quad i=1,2,3.$$ Later, Gurevich and Pitaevskii [@Gurevich; @1974] considered the problem of patching together solutions of Whitham’s modulational system with solutions of the formal limiting equation (obtained by setting $\epsilon=0$ in ) at two moving boundary points that delineate the oscillation zone; their goal was to provide a reasonable global approximation scheme for the solution of the initial-value problem for the KdV equation in the zero-dispersion limit subject to given initial data $v_\epsilon(x,0)=v_0(x)$ independent of $\epsilon$.

In the meantime, it was of course discovered that the KdV equation is a completely integrable system, posessing a compatible structure now called a *Lax pair* and a coincident solution procedure for addressing the Cauchy (initial-value) problem: the *inverse-scattering transform*. This development suggested that the methodology invented by Whitham could perhaps be placed on completely rigorous mathematical footing. After the exact periodic (and quasiperiodic) solutions of the KdV equation were given a spectral interpretation by Its and Matveev [@ItsM75] and Dubrovin, Matveev, and Novikov [@DubrovinMN76], the Whitham modulation equations themselves were reinterpreted within the framework of integrability by Flaschka, Forest, and McLaughlin [@FlaschkaFM80]. (In particular this 