Determining a nonlinear hyperbolic system with unknown sources and nonlinearity

TL;DR

This paper addresses inverse boundary problems for a time-dependent semilinear hyperbolic equation, demonstrating unique determination of unknown sources and nonlinearities through boundary observations using novel combined techniques.

Contribution

It introduces a new method combining observability and higher order linearization to simultaneously recover sources and nonlinearity in semilinear hyperbolic equations.

Findings

Unique determination of nonlinearity and sources in several scenarios

Development of a new technique combining observability and Runge approximation

Applicability to passive and active boundary observations

Abstract

We investigate inverse boundary problems associated with a time-dependent semilinear hyperbolic equation, where both nonlinearity and sources (including initial displacement and initial velocity) are unknown. We establish in several generic scenarios that one can uniquely determine the nonlinearity or the sources by using passive or active boundary observations. In order to exploit the nonlinearity and the sources simultaneously, we develop a new technique, which combines the observability for linear wave equations and a Runge approximation with higher order linearization for the semilinear hyperbolic equation.