Detection of Hermitian connections in wave equations with cubic non-linearity

TL;DR

This paper addresses the inverse problem of recovering a Hermitian connection in a cubic wave equation using microlocal analysis and a novel non-abelian broken light ray transform, with applications to Yang-Mills-Higgs equations.

Contribution

It introduces a new approach based on principal symbols and a non-abelian broken light ray transform to solve the inverse problem for nonlinear wave equations.

Findings

Successful recovery of Hermitian connections from source-to-solution maps

Development of a microlocal analysis method for nonlinear wave interactions

Introduction of a non-abelian broken light ray transform inversion technique

Abstract

We consider the geometric non-linear inverse problem of recovering a Hermitian connection $A$ from the source-to-solution map of the cubic wave equation $\Box_{A}\phi+\kappa |\phi|^{2}\phi=f$, where $\kappa\neq 0$ and $\Box_{A}$ is the connection wave operator in the Minkowski space $\mathbb{R}^{1+3}$. The equation arises naturally when considering the Yang-Mills-Higgs equations with Mexican hat type potentials. Our proof exploits the microlocal analysis of nonlinear wave interactions, but instead of employing information contained in the geometry of the wave front sets as in previous literature, we study the principal symbols of waves generated by suitable interactions. Moreover, our approach relies on inversion of a novel non-abelian broken light ray transform.