Electromagnetic pulse propagation in optical fibres is described by the non-linear Schrodinger equation. The solutions, or solitons, remain completely unchanged as they propagate along the fibre. The question we are concerned with is, given an initial input pulse, does it contain solitons, and if so, how many.
Answering this question means solving the non-linear Schrodinger equation and this is done by using the Inverse Scattering Method. This method utilises several linear problems which are compantively easier to solve - in this work we focus on the linear eigenvalue problem since it gives all the information about solitons.
In Chapter 1, we first show that pulse propagation in optical fibres is described by the non-linear Schrodinger equation. Chapter 2 deals with the Inverse Scattering method and, in particular, how it is used to solve the non-linear Schrodinger equation.
In Chapters 3, 4 and 5, the eigenvalue problem is exactly solved for three special families of input pulses. We show for the three cases that the soliton number depends upon the area of the pulse only, regardless of the pulse's shape.
Finally, in Chapter 6, the eigenvalue problem is discussed for the super-Gausslan pulse, the type of pulse produced by semiconductor lasers. Formal solutions are obtained in terms of an infinite series of functions. To calculate the exact solutions, numerical computations are required. We present the working software code and suggestions for tackling this problem.