At the equator, the Coriolis force from rotation vanishes
identically so that multiple time scale dynamics for the equatorial
shallow water equation naturally leads to singular limits of
symmetric hyperbolic systems with fast variable coefficients. The
classical strategy of using energy estimates for higher spatial
derivatives has a fundamental difficulty since formally the
commutator terms explode in the limit. Here this fundamental
difficulty is circumvented by exploiting the special structure of
the equatorial shallow water equations in suitable new variables
involving the raising and lowering operators for the quantum
harmonic oscillator, and obtaining uniform higher derivative
estimates in a new function space based on the Hermite operator. The
result is a completely new theorem characterizing the singular limit
of the equatorial shallow water equations in the long wave regime,
even with general unbalanced initial data, as a solution of the
equatorial long wave equation. The results presented below point the
way for rigorous PDE analysis of both the equatorial shallow water
equations and the equatorial primitive equations in other physically
relevant singular limit regimes.