Uniqueness For The Inverse Backscattering Problem Of Wave Equation

Wave equation is a kind of partial differential equation which is used to describe the fluctuation characteristic of sound wave, light wave, electromagnetic wave and so on. The mathematical theory research about the inverse backscattering problem of wave equation has always been the focus of the mathematics and physics, and we obtained some achievements about the uniqueness of sound speed. This paper mainly studies the inverse backscattering problem of wave equation, with the invariable potential. We mainly exploit the scattering kernel which is the backscattering echoes to study the uniqueness of the sound speed.The theory of uniqueness has always been important aspect in the study of partial differential equations.In this paper, we mainly use Lax-Phillips scattering theory to study the inverse backscattering problem of wave equation. And, we confirm the sound speed by measuring the response to waves in the backscattering direction. First, we introduce the scattering theory and obtain the relation of wave operator and scattering operator. The relationship between the scattering kernel and the scattering solution, the potential and the sound speed is obtained by the relation between the scattering operator and the modified Radon transform. Then we obtain the singular decomposition of the scattering solution by the eikonal equation and transport equations. And the properties of the coefficients in the expressions are obtained. Furthermore we turn the expression of scattering solution into the formula which we obtained. Due to the part containing the potential, in order to facilitate the estimation that we split the expression into two parts. Combined with the lemma which has introduced the proposition of operators and estimation of the coefficients in the expressions have been known, we receive the estimation of operators. Thus, the estimation of scattering kernel obtained. Finally, we obtain the estimation of sound speed by the relation of scattering kernel with sound speed. So we get the uniqueness of sound speed when the potential is invariant.