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

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 ${\square }_A\varphi +\kappa |\varphi {|}^2\varphi =f$, where $\kappa \ne 0$ and ${\square }_A$ 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.

Cited by in corpus (4)

• Uniqueness and stability of an inverse problem for a semi-linear wave equation
• Partial Data Inverse Problems for Nonlinear Magnetic Schrödinger Equations
• An Inverse Problem for Non-linear Fractional Magnetic Schrodinger Equation
• Recovering a Riemannian Metric from Cherenkov Radiation in Inhomogeneous Anisotropic Medium