Hamiltonian mechanics of generalized eikonal waves

TL;DR

This paper develops a Hamiltonian framework for generalized eikonal waves within the Keller-Maslov WKB theory, enabling structure-preserving numerical methods that are robust even in the presence of caustics.

Contribution

It introduces a Hamiltonian structure for the geometric semiclassical state in the Keller-Maslov framework, extending the theory beyond traditional wave ansatzes.

Findings

Hamiltonian, variational, and Poisson formulations for wave evolution

Insensitivity to caustics allows robust numerical integrators

Applicable to Hermitian wave operators in semiclassical analysis

Abstract

In accordance with the Keller-Maslov global WKB theory, a semiclassical scalar wave field is best encoded as a triple consisting of (i) a Lagrangian submanifold $\Lambda$ in the ray phase space, (ii) a density $\mu$ on $\Lambda$, and (iii) an overall phase factor $\phi$. We present the Hamiltonian structure of the Cauchy problem for such a "geometric semiclassical state" in the special case where the wave operator is Hermetian. Variational, symplectic, and Poisson formulations of the time evolution equations for $(\Lambda,\mu,\phi)$ are identitfied. Because we work in terms of the Keller-Maslov global WKB ansatz, as opposed to the more restrictive $\psi=a \exp(i S/\epsilon)$, all of our results are insensitive to the presence of caustics. In particular, because the variational principle is insensitive to caustics, the latter may be used to construct structure-perserving numerical integrators for scalar wave equations.