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Deconvolutional determination of the nonlinearity in a semilinear wave equation

Nicholas Hu, Rowan Killip, Monica Visan

TL;DR

The paper addresses identifying a quintic-type semilinear nonlinearity $F$ in the three-dimensional wave equation from scattering data. It develops a small-data scattering theory for admissible nonlinearities, derives Born-approximation asymptotics, and reduces the inverse problem to a deconvolution via carefully chosen linear solutions that concentrate at a spacetime point. Using Wiener’s Tauberian theorem and a nonvanishing Fourier transform for a weight function, it proves that the nonlinearity is uniquely determined at almost every spacetime point, even when $F$ depends on $t$ and $x$. This deconvolution framework provides a robust method for recovering nonlinearities in dispersive PDEs from scattering information, with potential applications to identifying nonlinear media shapes from wave propagation data.

Abstract

We demonstrate that in three space dimensions, the scattering behaviour of semilinear wave equations with quintic-type nonlinearities uniquely determines the nonlinearity. The nonlinearity is permitted to depend on both space and time.

Deconvolutional determination of the nonlinearity in a semilinear wave equation

TL;DR

The paper addresses identifying a quintic-type semilinear nonlinearity in the three-dimensional wave equation from scattering data. It develops a small-data scattering theory for admissible nonlinearities, derives Born-approximation asymptotics, and reduces the inverse problem to a deconvolution via carefully chosen linear solutions that concentrate at a spacetime point. Using Wiener’s Tauberian theorem and a nonvanishing Fourier transform for a weight function, it proves that the nonlinearity is uniquely determined at almost every spacetime point, even when depends on and . This deconvolution framework provides a robust method for recovering nonlinearities in dispersive PDEs from scattering information, with potential applications to identifying nonlinear media shapes from wave propagation data.

Abstract

We demonstrate that in three space dimensions, the scattering behaviour of semilinear wave equations with quintic-type nonlinearities uniquely determines the nonlinearity. The nonlinearity is permitted to depend on both space and time.
Paper Structure (5 sections, 9 theorems, 73 equations)

This paper contains 5 sections, 9 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Small-data scattering
  4. Reduction to a convolution equation
  5. Deconvolutional determination of the nonlinearity

Key Result

Theorem 1.3

Let $F$ be an admissible nonlinearity for nlw. Then there exists an $\eta > 0$ such that nlw has a unique global solution $u$ satisfying whenever $(u_0, u_1) \in B_\eta$, where This solution scatters in $\dot{H}^1(\mathbb{R}^3) \times L^2(\mathbb{R}^3)$ as $t \to \pm \infty$, meaning that there exist (necessarily unique) asymptotic states $(u^\pm_0, u^\pm_1) \in \dot{H}^1(\mathbb{R}^3) \times L^

Theorems & Definitions (19)

  • Definition 1.1: Admissible nonlinearity
  • Definition 1.2: Solution
  • Theorem 1.3: Small-data scattering
  • Definition 1.4: Determinable point
  • Theorem 1.5
  • Theorem 2.1: Strichartz estimates, PecherStrichartzKeel
  • proof : Proof of \ref{['T:scattering']}
  • Corollary 2.2: Small-data asymptotics for the wave and scattering operators
  • proof
  • Lemma 3.1
  • ...and 9 more