Exact solution of the initial eigenvalue problem for coherent pulse propagation

Abstract

The inverse scattering method is used to investigate the resonant propagation of an optical pulse assumed to be initially unchirped and to have a hyperbolic-secant shape. An exact solution is found for the two-component initial eigenvalue problem. The discrete spectrum consists of a finite set of purely imaginary, equidistant values. The number of discrete eigenvalues (number of solitons) is determined by the initial pulse area $A$. The scattering amplitudes are expressed in terms of Euler gamma functions and hyperbolic functions. In addition to poles in the upper half-plane, the function $\frac{\overline{b}}{a}$ has a pole on the real axis for $A=(2n+1)\ensuremath{\pi}$.