Inverse scattering problems for non-linear wave equations on Lorentzian manifolds

TL;DR

This paper demonstrates that for semilinear wave equations on certain Lorentzian manifolds, the scattering data uniquely determine the manifold's geometric and topological structure, even allowing for cases with blow-up solutions.

Contribution

It introduces non-linear scattering functionals to solve inverse problems on complex manifolds, extending inverse scattering theory to non-linear and topologically intricate settings.

Findings

Scattering functionals determine manifold topology and conformal class.

The metric and non-linearity coefficient are uniquely identified up to a multiplicative factor.

The method applies even when classical scattering operators are undefined due to blow-up.

Abstract

We show that an inverse scattering problem for a semilinear wave equation can be solved on a manifold having an asymptotically Minkowskian infinity, that is, scattering functionals determine the topology, differentiable structure, and the conformal type of the manifold. Moreover, the metric and the coefficient of the non-linearity are determined up to a multiplicative transformation. The manifold on which the inverse problem is considered is allowed to be an open, globally hyperbolic manifold which may have non-trivial topology or several infinities (i.e., ends) of which at least one has to be of the asymptotically Minkowskian type. To formulate the inverse problems we define a new type of data, non-linear scattering functionals, which are defined also in the cases where the classically defined scattering operators are not well-defined. This makes it possible to solve inverse problems also in cases where some of the incoming waves lead to a blow-up of the scattered solution. We use non-linear interaction of waves as a beneficial tool that helps to solve the inverse problem. The corresponding inverse problem for the linear wave equation still remains unsolved.