Rigorous justification of the geometric-optics approximation

Prove a rigorous justification of the geometric-optics approximation for the noise correlation modes governing the unresolved wave dynamics.

Background

The paper derives WKB and geometric-optics descriptions for the noise correlation modes. In this approximation, the modes satisfy a Hamilton–Jacobi equation with a generally complex dispersion relation, and their wavevectors and positions evolve along associated bicharacteristics.

The derivation is asymptotic and formal: it assumes slowly varying amplitudes and phases and extracts leading-order behavior from the transport equation. The paper does not establish error bounds, convergence, or validity over specified time intervals, and explicitly leaves the rigorous justification of this approximation unresolved.

References

From a mathematical standpoint, the well-posedness of the coupled system at vanishing correlation time, and the rigorous justification of the geometric-optics approximation remain open questions.

— Coupled stochastic variational principles for multiscale surface gravity waves -- Part I: theoretical framework  (2608.27074 - Mémin et al., 27 Aug 2026) in Conclusion; Section 6, “Geometric-optics solution for the correlation modes”