Reconstruction of potential and damping coefficients in a semi-linear wave equation

TL;DR

The paper addresses the inverse problem of determining the damping $a$, linear potential $b$, and nonlinear potential $q$ in a semi-linear wave equation from boundary measurements encoded by the Dirichlet-to-Neumann map $\Lambda$. It employs a higher-order linearization strategy, constructing geometric-optics and asymptotic solutions to extract integral identities that reveal the unknown coefficients. The authors first recover $a$ and $b$ via the first-order linearization and ray transforms, then determine $q$ through second-order linearization and an asymptotic analysis that yields a transport equation for $q$ along lines. The results provide a rigorous well-posed forward theory and a constructive reconstruction procedure in the Euclidean setting, extending prior work on nonlinear inverse problems to include damping and linear potentials with boundary data. This advances noninvasive characterization of media exhibiting damping, stiffness variation, and nonlinear responses, with potential applications in geophysics and acoustics.

Abstract

In this article, we investigate an inverse problem for a semi-linear wave equation posed in $\mathbb{R}^{n+1}$, with $n \geq 2$. Our primary objective is to reconstruct the damping coefficient, the linear potential, and the nonlinear potential from the associated Dirichlet-to-Neumann map. The analysis is based on a higher-order linearization method. As a key step, we establish the existence of suitable asymptotic solutions, crucial for reconstructing the nonlinear potential. In addition, we also provide a detailed study of the corresponding forward problem.

Theorems & Definitions (11)

• Theorem 1.1
• Lemma 2.1: Well-posedness for linear equations with potential (lin2024determining Lemma 3.2)
• Lemma 2.2: Well-posedness for linear equations with potential and damping terms
• proof
• Theorem 2.3
• proof
• Lemma 3.1
• Lemma 3.2
• Proposition 3.3: Asymptotic solutions
• Lemma 3.4