Inverse problems for nonlinear progressive waves

TL;DR

This paper investigates inverse problems for nonlinear progressive waves, establishing unique identifiability of wave parameters and initial data using boundary measurements, with implications for infrasound waveform inversion.

Contribution

It introduces new theoretical results on the unique recovery of nonlinear wave equations and initial data from boundary data, employing high-order linearisation and Gaussian beam solutions.

Findings

Proves unique identifiability of the nonlinear wave function and initial data.

Develops methods using high-order linearisation and Gaussian beams.

Connects theoretical results to practical infrasound waveform inversion applications.

Abstract

We propose and study several inverse problems associated with the nonlinear progressive waves that arise in infrasonic inversions. The nonlinear progressive equation (NPE) is of a quasilinear form $\partial_t^2 u=\Delta f(x, u)$ with $f(x, u)=c_1(x) u+c_2(x) u^n$, $n\geq 2$, and can be derived from the hyperbolic system of conservation laws associated with the Euler equations. We establish unique identifiability results in determining $f(x, u)$ as well as the associated initial data by the boundary measurement. Our analysis relies on high-order linearisation and construction of proper Gaussian beam solutions for the underlying wave equations. In addition to its theoretical interest, we connect our study to applications of practical importance in infrasound waveform inversion.