Pulse propagation in a nonlinear optical fibre of parabolic index profile by direct numerical solution of the Gel´fand-Levitan integral equations

The evolution of an optical pulse in a graded-index (parabolic profile) single-mode nonlinear optical fibre is treated by means of differential equation techniques. Using a slowly varying envelope approximation and an averaging method over the transverse direction, a differential equation of the nonlinear Schrodinger type is obtained for the unknown envelope function of the electric field. The inverse scattering method is then applied, leading to the equivalent system of Gel´fand-Levitan-Marchenko coupled integral equations. A new iterative solution to these equations is presented. As an example, numerical results for a hyperbolic secant initial pulse profile of variable amplitude are obtained and the behaviour of both the soliton and the radiation part of the solution is examined. Finally, in the special case of reflectionless potentials, the well known analytical single and double solitons are recovered