Pathfollowing for essentially singular boundary value problems with application to the complex Ginzburg-Landau equation

Abstract

We present a pathfollowing strategy based on pseudo-arclength parametrization for the solution of parameter-dependent boundary value problems for ordinary differential equations. We formulate criteria which ensure the successful application of this method for the computation of solution branches with turning points for problems with an essential singularity. The advantages of our approach result from the possibility to use efficient mesh selection, and a favorable conditioning even for problems posed on a semi-infinite interval and subsequently transformed to an essentially singular problem. This is demonstrated by a Matlab implementation of the solution method based on an adaptive collocation scheme which is well suited to solve problems of practical relevance. As one example, we compute solution branches for the complex Ginzburg-Landau equation which start from non-monotone ‘multi-bump’ solutions of the nonlinear Schrödinger equation. Following the branches around turning points, real-valued solutions of the nonlinear Schrödinger equation can easily be computed.