  It is proved that approximations which are obtained as solutions of the
multiphase Whitham modulation equations stay close to solutions of the original
equation on a natural time scale. The class of nonlinear wave equations chosen
for the starting point is coupled nonlinear Schr\"odinger equations. These
equations are not in general integrable, but they have an explicit family of
multiphase wavetrains that generate multiphase Whitham equations which may be
elliptic,hyperbolic, or of mixed type. Due to the change of type, the function
space setup is based on Gevrey spaces with initial data analytic in a strip in
the complex plane. In these spaces a Cauchy- Kowalevskaya-like existence and
uniqueness theorem is proved. Building on this theorem and higher-order
approximations to Whitham theory, a rigorous comparison of solutions, of the
coupled nonlinear Schr\"odinger equations and the multiphase Whitham modulation
equations, is obtained.
