Detection of Hermitian connections in wave equations with cubic non-linearity
Detection of Hermitian connections in wave equations with cubic non-linearity
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DOI：
10.4171/jems/1136
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发表时间：
2019-02
影响因子：
2.6
通讯作者：
Xi Chen;M. Lassas;L. Oksanen;G. Paternain
中科院分区：
文献类型：
--
作者：
Xi Chen;M. Lassas;L. Oksanen;G. Paternain
考虑了从三次波动方程$\Box_{A}\phi+\kappa的源到解映射恢复厄米特联络$A$的几何非线性反问题|\phi| ^{2}\phi=f$，其中$\kappa\neq 0$和$\Box_{A}$是Minkowski空间$\mathbb{R}^{1+3}$中的联络波算子。当考虑具有墨西哥帽型势的Yang-Mills-Higgs方程时，方程自然出现。我们的证明利用了非线性波相互作用的微局部分析，但不是采用信息中包含的几何形状的波前集在以前的文献中，我们研究的主要符号的波所产生的适当的相互作用。此外，我们的方法依赖于一种新的非阿贝尔破碎光线变换的反演。
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.