Stability Analysis of Pulses via the Evans Function: Dissipative Systems

Linear stability analysis of pulses is considered in this review chapter. The Evans function is an analytic tool whose zeros correspond to eigenvalues. Herein, the general manner of its construction shown. Furthermore, the construction is done explicitly for the linearization of the nonlinear Schrödinger equation about the 1-soliton solution. In an explicit calculation, it is shown how the Evans function can be used to track the non-zero eigenvalues arising from a dissipative perturbation of the nonlinear Schrödinger equation which arises in the context of pulse propagation in nonlinear optical fibers.