Inverse Boundary Value Problem for Non-linear Hyperbolic Partial Differential Equations

TL;DR

This paper investigates an inverse boundary value problem for a non-linear wave equation in higher dimensions, demonstrating that boundary measurements can uniquely determine the spatially varying wave speed and quadratic coefficients of the non-linearity.

Contribution

It introduces a method to uniquely recover the wave speed and quadratic non-linear coefficients from boundary data in a non-linear hyperbolic PDE setting.

Findings

Unique determination of b3(x) and quadratic coefficients from boundary measurements.

Linearization at trivial solution reduces to a linear wave equation with perturbations.

Boundary measurements over finite time suffice for parameter recovery.

Abstract

In this article we are concerned with an inverse boundary value problem for a non-linear wave equation of divergence form with space dimension $n\geq 3$. In particular the so called the interior determination problem. This non-linear wave equation has a trivial solution, i.e. zero solution. By linearizing this equation at the trivial solution, we have the usual linear isotropic wave equation with the speed $\sqrt{\gamma(x)}$ at each point $x$ in a given spacial domain. For any small solution $u=u(t,x)$ of this non-linear equation, we have the linear isotropic wave equation perturbed by a divergence with respect to $x$ of a vector whose components are quadratics with respect to $\nabla_x u(t,x)$ by ignoring the terms with smallness $O(|\nabla_x u(t,x)|^3)$. We will show that we can uniquely determine $\gamma(x)$ and the coefficients of these quadratics by many boundary measurements at the boundary of the spacial domain over finite time interval. More precisely the boundary measurements are given as the so-called the hyperbolic Dirichlet to Neumann map.