Wiggling Ward's Waves with Miura's Metaplectic Magic

This talk continues a series on the geometry and analysis of gravitational plane waves. The earlier talks described a fixed plane wave in three complementary ways: as a curve in the Lagrangian Grassmannian of transverse Jacobi fields, as a Schrödinger system obtained by Fourier reduction, and as a Ward-type propagator whose quadratic phase depends on a choice of polarization. The present talk asks a new question: how should plane-wave data deform?

I will explain why the natural deformation problem is not arbitrary perturbation theory, but zero-curvature deformation of the symplectic first-order system underlying the Jacobi equation. The first nontrivial odd deformation is the matrix KdV equation. I will then show how the Miura transformation arises geometrically from plane-wave data itself: a chosen polarization, or Rosen frame, determines a first-order Riccati variable, and the Miura map is simply the passage from this polarized first-order data to the invariant Brinkmann curvature profile. From this viewpoint, Miura is not an ad hoc substitution but the local expression of the same measurement structure that governed the Ward propagator in the previous talk.

I will also discuss a simple 2×2 rotating homogeneous example, and conclude with the physical meaning of the deformation. In the periodic finite-gap regime, matrix KdV preserves the Bloch spectrum of the associated wave operator while canonically deforming the quadratic phase of the Ward propagator. Thus the flow preserves the spectral content of the wave while changing the focusing data seen by a chosen polarization.