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The Recovery of Semilinear Potentials Satisfying Null Conditions From Scattering Data

Joel Nathe, Antônio Sá Barreto

TL;DR

The article addresses the inverse problem of recovering a semilinear potential under a null condition from scattering-type data for a semilinear wave equation $\square u= q(x,u)((\partial_{x_0}u)^2-|\nabla_{x'}u|^2)$. It develops a nonlinear geometric optics framework to construct highly oscillatory solutions on finite time intervals, linking the amplitude of oscillations to the light-ray transform of a vector field derived from $q$. By translating approximate nonlinear solutions into genuine ones through Gu\`es-type energy estimates in semiclassical Sobolev spaces, the authors establish uniqueness of the nonlinear potential $q$ in the region determined by the data. The work provides a rigorous route from oscillatory constructions to a concrete inverse result for semilinear wave equations with null forms, contributing a robust tool for nonlinear inverse problems and scattering-type analysis.

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.

The Recovery of Semilinear Potentials Satisfying Null Conditions From Scattering Data

TL;DR

The article addresses the inverse problem of recovering a semilinear potential under a null condition from scattering-type data for a semilinear wave equation . It develops a nonlinear geometric optics framework to construct highly oscillatory solutions on finite time intervals, linking the amplitude of oscillations to the light-ray transform of a vector field derived from . By translating approximate nonlinear solutions into genuine ones through Gu\`es-type energy estimates in semiclassical Sobolev spaces, the authors establish uniqueness of the nonlinear potential in the region determined by the data. The work provides a rigorous route from oscillatory constructions to a concrete inverse result for semilinear wave equations with null forms, contributing a robust tool for nonlinear inverse problems and scattering-type analysis.

Abstract

We construct oscillatory solutions of fully semilinear wave equations in Minkowski space satisfying a null condition of the form on an interval , arbitrary, which consist of the superposition of a non-oscillatory background solution and a single phase train of highly oscillatory waves of wave length and amplitudes given by powers of ; the waves interact with the nonlinearity and we measure the response at a fixed time . We show that the coefficient of amplitude of the oscillatory part of the nonlinear geometric optics expansion of the solution determines the light-ray transform of a vector field associated with , which determines 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.
Paper Structure (7 sections, 15 theorems, 114 equations)

This paper contains 7 sections, 15 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. The Inverse Problem
  3. Approximate Solutions of Equation \ref{['NLWE']}
  4. From Approximate Solutions to Solutions of \ref{['NLWE']}
  5. Spaces of Distributions
  6. Energy Estimates
  7. The Proof of Theorem \ref{['approx']}

Key Result

Theorem 1.1

Let $q(x,u)$ be supported on $\{|x|\leq R\} \times {\mathbb R}$. Let $\varphi , \chi\in C_0^\infty({\mathbb R})$ be real valued, let $\omega, \theta\in {\mathbb S}^{n-1}$, $V=(\pm 1,\theta)$ and $W=( -1,\omega)$. Let $\varphi_{{}_V}(x)$ and $\chi_{{}_W}(x)$ be defined as in defphi. For $A,B\in {\mat which satisfies NLWE if $x$ is not on the support of $\chi_{{}_W}$. For any $N\in {\mathbb N}$ and

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • proof
  • Proposition 1.3
  • proof
  • Theorem 2.1
  • proof
  • Theorem 3.1
  • Definition 3.2
  • Lemma 3.3
  • ...and 16 more