Mathematical Formulation of Inverse Scattering and Korteweg- De Vries Equation

Inverse scattering refers to the determination of the solutions of a set of differential equations based on known asymptotic solutions, that is, the solution of Marchenko equation. Marchenko equation was derived using integral equation. The potential function derived from eigenvalues and scattering data seems to be the inverse method of scattering problem. The reflection coefficient with one pole and zero reflection coefficients has been chosen to solve inverse scattering problem. Again this paper deals with the connection between inverse scattering and the Korteweg-de Vries equation and describes variety of examples with Korteweg-de Vries equation: the single-soliton solution, the two-soliton solution and finally the N-soliton solution. Throughout the work, the primary objective is to study some mathematical techniques applied in analyzing the behavior of soliton in the KdV equations. Keywords: Marchenko equation, KdV equation, Solitons, Scattering, Inverse Scattering, Canal.