Inverse spectral theory is used to prescribe and study equations for the slow modulations of
N‐phase wave trains for the Korteweg‐de Vries (KdV) equation. An invariant representation
of the modulational equations is deduced. This representation depends upon certain
differentials on a Riemann surface. When evaluated near∞ on the surface, the invariant
representation reduces to averaged conservations laws; when evaluated near the branch
points, the representation shows that the simple eigenvalues provide Riemann invariants for …