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.
"The dynamics of equatorial long waves: a singular limit with fast variable coefficients." Commun. Math. Sci. 4 (2) 375 - 397, June 2006.