Existence, uniqueness, and numerical solutions of the nonlinear periodic Westervelt equation

TL;DR

This paper investigates the nonlinear periodic Westervelt equation, establishing existence, uniqueness, and proposing a numerical method for solving the system, with applications in improving ultrasound imaging techniques.

Contribution

It provides the first comprehensive analysis of existence, uniqueness, and a multiharmonic numerical scheme for the nonlinear Westervelt equation with variable coefficients.

Findings

Higher harmonics are generated through nonlinear wave propagation.

The iterative scheme converges under small excitation conditions.

Finite element method effectively discretizes the Helmholtz system.

Abstract

In this paper, we study the nonlinear periodic Westervelt equation with excitations located within a bounded domain in $\mathbb{R}^d$, where $d \in \{2,3\}$, subject to Robin boundary conditions. This problem is of particular interest for advancing imaging techniques that exploit nonlinearity of the acoustic propagation. We establish the existence and uniqueness of solutions in both the linear and the nonlinear setting, thereby allowing for spatially varying coefficients as relevant in quantitative imaging. Derivation of a multiharmonic formulation enables us to show the generation of higher harmonics (that is, responses at multiples of the fundamental frequency) due nonlinear wave propagation. An iterative scheme for solving the resulting system is proposed that relies on successive resolution of these higher harmonics, and its convergence under smallness conditions on the excitation is proven. Furthermore, we investigate the numerical solution of the resulting system of Helmholtz equations, employing a conforming finite element method for its discretization. Through an implementation of the proposed methodology, we illustrate how acoustic waves propagate in nonlinear media. This study aims to enhance our understanding of ultrasound propagation dynamics, which is essential for obtaining high-quality images from limited in vivo and boundary measurements.