The Recovery of Semilinear Potentials Satisfying Null Conditions From Scattering Data

TL;DR

This paper develops a method to recover the semilinear potential function in wave equations satisfying null conditions from scattering data, using oscillatory solutions and geometric optics expansions.

Contribution

It introduces a novel approach to determine the nonlinear potential from boundary measurements by analyzing high-frequency oscillatory solutions.

Findings

The amplitude of oscillatory solutions encodes the light-ray transform of the potential.

The potential $q(x,u)$ can be uniquely recovered in the maximal region from scattering data.

Methods extend to systems of semilinear wave equations with null conditions.

Abstract

We construct oscillatory solutions of fully semilinear wave equations in Minkowski space satisfying a null condition of the form $$\square u:=(-\partial_{x_0}^2 +\sum_{j=1}^n \partial_{x_j}^2 )u= q(x,u)((\partial_{x_0}u)^2-|\nabla_{x'}u|^2),$$ $$x=(x_0,x'), \;\ x'=(x_1,\ldots, x_n) \text{ and } x_0=t \text{ is the time variable,}$$ on an interval $x_0\in [-T,T]$, $T<\infty$ arbitrary, which consist of the superposition of a non-oscillatory background solution and a single phase train of highly oscillatory waves of wave length $h\ll1$ and amplitudes given by powers of $h$; the waves interact with the nonlinearity and we measure the response $u(x_0,x')|_{x_0=T'}$ at a fixed time $x_0=T'<T$. We show that the coefficient of amplitude $h$ of the oscillatory part of the nonlinear geometric optics expansion of the solution determines the light-ray transform of a vector field associated with $q(x,u)$, which determines $q(x,u)$ uniquely in the maximal region determined by the data. Our methods also work for systems of semilinear wave equations satisfying null conditions, but in this paper we focus on the scalar case.