Inverse Scattering Problems for Hyperbolic Equations and Their Applications

Abstract

For the wave equation

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbWexLMBbXgBd9gzLbvyNv2CaeHbl7mZLdGeaGqiVu0Je9sqqr % pepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs % 0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaai % aabeqaamaabaabauaakeaacaWGmbGaeqiYdKNaeyyyIO7aaSaaaeaa % cqGHciITdaahaaWcbeqaaiaaikdaaaGccqaHipqEaeaacqGHciITca % WG4bGaeyOaIyRaamyEaaaacqGHRaWkcaWG1bGaaiikaiaadIhacaGG % SaGaamyEaiaacMcacqaHipqEcqGH9aqpcaaIWaGaaiilaiaadwhacq % GHiiIZcaWGmbWaaSbaaSqaaiaaigdaaeqaaOGaaiikaiaadweadaah % aaWcbeqaaiaaikdaaaGccaGGPaaaaa!5DF9!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$L\psi \equiv \frac{{{\partial ^2}\psi }}{{\partial x\partial y}} + u(x,y)\psi = 0,u \in {L_1}({E^2})$$

(1)

the Riemann function r(x, y; ξ, η) (the solution of the equation (1) which is equal to 1 along the characteristics x = ξ and y = η) is uniquely determined by the scattering data T(x,y) = r(+∞, y; x,−∞)−1. Therefore the potential u(x, y) is uniquely reconstructed by the scattering data. The basis of the algorithm for the solving such an inverse problem is the integral equations of the Gelfand-Levitan-Marchenko type. They are of the form: