TY - JOUR

T1 - The Problem of Determining the Coefficient of the Nonlinear Term in a Quasilinear Wave Equation

AU - Romanov, V. G.

AU - Bugueva, T. V.

N1 - Funding Information: The work was carried out within the framework of the state assignment for Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences, project no. FWNF-2022-0009. Publisher Copyright: © 2022, Pleiades Publishing, Ltd.

PY - 2022/5

Y1 - 2022/5

N2 - For a nonlinear differential equation whose main part is the wave operator, we considerthe inverse problem of determining the coefficient of the nonlinear term in the equation. It isassumed that the desired coefficient is a continuous compactly supported function in R3. For the original equation, we consider plane waves incident on theinhomogeneity at different angles. In the inverse problem, it is assumed that the solutionscorresponding to these waves can be measured at points on the boundary of a certain ballcontaining the inhomogeneity at times close to the wave front arrival at these points and for acertain range of angles of incidence of the plane wave. It is shown that the solutions of thecorresponding direct problems for the differential equation are bounded in some neighborhood ofthe wave front, and an asymptotic expansion of the solution is found in this neighborhood. On thebasis of this expansion, it is established that the information specified in the inverse problemallows reducing the problem of finding the desired function to the problem of X-ray tomographywith incomplete data. A theorem on the uniqueness of the solution of the inverse problem isstated and proved. It is shown that in an algorithmic sense, this problem is reduced to thewell-known moment problem.

AB - For a nonlinear differential equation whose main part is the wave operator, we considerthe inverse problem of determining the coefficient of the nonlinear term in the equation. It isassumed that the desired coefficient is a continuous compactly supported function in R3. For the original equation, we consider plane waves incident on theinhomogeneity at different angles. In the inverse problem, it is assumed that the solutionscorresponding to these waves can be measured at points on the boundary of a certain ballcontaining the inhomogeneity at times close to the wave front arrival at these points and for acertain range of angles of incidence of the plane wave. It is shown that the solutions of thecorresponding direct problems for the differential equation are bounded in some neighborhood ofthe wave front, and an asymptotic expansion of the solution is found in this neighborhood. On thebasis of this expansion, it is established that the information specified in the inverse problemallows reducing the problem of finding the desired function to the problem of X-ray tomographywith incomplete data. A theorem on the uniqueness of the solution of the inverse problem isstated and proved. It is shown that in an algorithmic sense, this problem is reduced to thewell-known moment problem.

KW - inverse problem

KW - nonlinear wave equation

KW - tomography

UR - http://www.scopus.com/inward/record.url?scp=85144179394&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/e599bc98-f7a3-3171-a30a-c5c28b45a01d/

U2 - 10.1134/S1990478922030188

DO - 10.1134/S1990478922030188

M3 - Article

AN - SCOPUS:85144179394

VL - 16

SP - 550

EP - 562

JO - Journal of Applied and Industrial Mathematics

JF - Journal of Applied and Industrial Mathematics

SN - 1990-4789

IS - 3

ER -