Unique determination of sound speeds for coupled systems of semi-linear wave equations

• Published in: Indagationes mathematicae Vol. 30; no. 5; pp. 904 - 919
• Main Author: Waters, Alden
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.09.2019; Elsevier Science Ltd
• Subjects: Coupled systems; Inverse problems; Mathematical analysis; Non-linear hyperbolic equations; Nonlinear equations; Sound; Wave equations; Coupled systems; Inverse problems; Non-linear hyperbolic equations
• ISSN: 0019-3577; 1872-6100
• DOI: 10.1016/j.indag.2019.07.003

We consider coupled systems of semi-linear wave equations with different sound speeds on a finite time interval [0,T] and a bounded domain Ω in R3 with C1 boundary ∂Ω. We show the coupled systems are well posed for variable coefficient sound speeds and short times. Under the assumption of small initial data, we prove the source to solution map associated with the nonlinear problem is sufficient to determine the source to solution map for the linear problem. This result is a bit surprising because one does not expect, in general, for the interaction of the waves in the nonlinear problem to always behave in a tractable fashion. As a result, we can reconstruct the sound speeds in Ω for the coupled nonlinear wave equations under certain geometric assumptions. In the case of the full source to solution map in Ω×[0,T] this reconstruction could also be accomplished under fewer geometric assumptions.