JournalsjemsVol. 24, No. 7pp. 2191–2232

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

  • Xi Chen

    Fudan University, Shanghai, China
  • Matti Lassas

    University of Helsinki, Finland
  • Lauri Oksanen

    University of Helsinki, Finland
  • Gabriel P. Paternain

    University of Cambridge, UK
Detection of Hermitian connections in wave equations with cubic non-linearity cover
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Abstract

We consider the geometric non-linear inverse problem of recovering a Hermitian connection AA from the source-to-solution map of the cubic wave equation Aϕ+κϕ2ϕ=f\Box_{A}\phi+\kappa |\phi|^{2}\phi=f, where κ0\kappa\neq 0 and A\Box_{A} is the connection wave operator in the Minkowski space R1+3\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 non-linear 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, a result interesting in its own right.

Cite this article

Xi Chen, Matti Lassas, Lauri Oksanen, Gabriel P. Paternain, Detection of Hermitian connections in wave equations with cubic non-linearity. J. Eur. Math. Soc. 24 (2022), no. 7, pp. 2191–2232

DOI 10.4171/JEMS/1136