Recovering coefficients in a nonlinear acoustics equation for an inhomogeneous media

Abstract A nonlinear acoustic equation with a variable sound speed <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>c</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>𝐱</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {c(\mathbf{x})} is considered. This equation contains an unknown coefficient <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>q</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>𝐱</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {q(\mathbf{x})} under a nonlinearity that plays the important role in the medical diagnostics. Relating to this, in this article a problem of recovering this coefficient is studied. Simultaneously, the problem of the sound speed recovering is also considered. To find the both coefficients, an information about solutions of a direct problem for the original equation is given. It is supposed that the sound speed differs from a given constant <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>c</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>&gt;</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {c_{0}&gt;0} in the ball B and coefficient <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>q</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>𝐱</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {q(\mathbf{x})} is compactly supported with the support in B . A plane wave propagates from the infinity in the direction <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ν</m:mi> <m:mn>0</m:mn> </m:msup> </m:math> {\nu^{0}} and it is registered on the boundary of ball B for different <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ν</m:mi> <m:mn>0</m:mn> </m:msup> </m:math> {\nu^{0}} . This information lies in the basement of the inverse problem. The structure of the solution to the direct problem is studied. As a result, the inverse problem of recovering the sound speed is reduced to the well-known inverse kinematic problem and the problem of recovering <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>q</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>𝐱</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {q(\mathbf{x})} is reduced to a problem of the integral geometry for a family of geodesic lines of the Riemannian metric with a given weight function. The latter problem is new and it is studied in this article. The uniqueness and stability theorems for the problem of the integral geometry and for the inverse problem are stated.