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The paper studies inverse problems of determining unknown coefficients invarious semi-linear and quasi-linear wave equations. We introduce a method tosolve inverse problems for non-linear equations using interaction of threewaves, that makes it possible to study the inverse problem in all dimensions$n+1\geq 3$. We consider the case when the set $\Omega_{\textrm{in}}$, wherethe sources are supported, and the set $\Omega_{\textrm{out}}$, where theobservations are made, are separated. As model problems we study both aquasi-linear and also a semi-linear wave equation and show in each case that itis possible to uniquely recover the background metric up to the naturalobstructions for uniqueness that is governed by finite speed of propagation forthe wave equation and a gauge corresponding to change of coordinates. The proofconsists of two independent components. In the first half we studymultiple-fold linearization of the non-linear wave equation near real parts ofGaussian beams that results in a three-wave interaction. We show that thethree-wave interaction can produce a three-to-one scattering data. In thesecond half of the paper, we study an abstract formulation of the three-to-onescattering relation showing that it recovers the topological, differential andconformal structures of the manifold in a causal diamond set that is theintersection of the future of the point $p_{in}\in \Omega_{\textrm{in}}$ andthe past of the point $p_{out}\in \Omega_{\textrm{out}}$. The results do notrequire any assumptions on the conjugate or cut points.

Inverse problems for non-linear hyperbolic equations with disjoint sources and receivers