The problem of determining the coefficient for a nonlinear term of a quasi-linear wave equation

V. G. Romanov ; T.V. Bugueva

Sibirskij žurnal industrialʹnoj matematiki, Tome 25 (2022) no. 3, pp. 154-169

Résumé

For a nonlinear differential equation, the main part of which is the wave operator, we consider the inverse problem of determining the coefficient of the nonlinear term of the equation. It is assumed that the desired coefficient is a continuous and finite function in $\mathbb{R}^3$. For the original equation, plane waves are considered that are fall on the inhomogeneity at different angles. In the inverse problem, it is assumed that the solutions corresponding to these waves can be measured at the points at the boundary of a certain ball containing the inhomogeneity, at times close to the arrival of the wave front at these points, and for a certain range of angles of incidence of plane. It is shown that the solutions of the corresponding direct problems for the differential equation are bounded in some neighborhood of the wave front, and an asymptotic expansion of the solution in this neighborhood is found. On the basis of this expansion, it was established that the information specified in the inverse problem allows us to reduce the problem of finding the desired function to the problem of X-ray tomography with incomplete data. A theorem on the uniqueness of the solution of the inverse problem is formulated and proved. It is shown that, in an algorithmic sense, this problem is reduced to the well-known problem of moments.

Détail

Keywords: nonlinear wave equation, inverse problem, tomography. .