Weakly nonlinear geometric optics for the Westervelt equation and recovery of the nonlinearity

TL;DR

This paper investigates the weakly nonlinear Westervelt equation, deriving a Burgers' type profile equation and demonstrating how to reconstruct the nonlinearity from high-frequency wave measurements using X-ray transform techniques.

Contribution

It introduces a novel approach to recover the nonlinearity in the Westervelt equation via wave packet tilts and X-ray transform, linking nonlinear wave behavior to inverse problem methods.

Findings

The leading profile equation is of Burgers' type.

Nonlinearity can be reconstructed from wave packet tilts.

Reconstruction uses the X-ray transform of the nonlinearity.

Abstract

We study the non-diffusive Westervelt equation in the weakly nonlinear regime. We show that the leading profile equation is of Burgers' type. We show that a compactly supported nonlinearity $\alpha$ can be reconstructed from the tilt of the transmitted high frequency wave packets sent from different directions since those tilts are proportional to the X-ray transform of $\alpha$.