Assessing Mathematical Forms employed to Describe Evolutions of Perturbations Propagating inside Optical Fibers
Comparisons have been made with different mathematical forms used to represent perturbation fields in studying the modulation instability happened in optical fibers. Without an artificial constraint, self consistent solutions to the linearized nonlinear Schrodinger equation can be deduced, which has recovered a rich variety of evolution characteristics at the initial stage for the perturbations co-propagating with a strong CW radiation inside the optical fiber. Although previous theories have correctly predicted the perturbation gain coefficient, an effort of seeking a constant gain coefficient as suggested in literatures has directly focused the discussions to the asymptotic characteristics of perturbations and therefore failed to give a comprehensive description to the evolutions.
Given ?>0 andp?(0,1), we consider the following problem: findu such that $$\begin{gathered} - \Delta u + \lambda [u]_ + ^p = 0in\Omega , \hfill \\ u = 1on\partial \Omega , \hfill \\ \end{gathered} $$ whereΩ??2 is a smooth convex domain. We prove optimalH 1 andL ? error bounds for the standard continuous piecewise linear Galerkin finite element approximation. In addition we analyse a more practical approximation using numerical integration on the nonlinear term. Finally we consider a modified nonlinear SOR algorithm, which is shown to be globally convergent, for solving the algebraic system derived from the more practical approximation.