On self-similar singular solutions of the complex Ginzburg-Landau equation

Petr Plechac and Vladimír Šverák

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Submission date: 10. Aug. 2000
Pages: 25
published in: Communications on pure and applied mathematics, 54 (2001) 10, p. 1215-1242 
DOI number (of the published article): 10.1002/cpa.3006
Bibtex
MSC-Numbers: 35Q55
Keywords and phrases: ginzburg-landau equation, nonlinear schrödinger equation, singularities

Abstract:
We address the open problem of existence of singularities for the complex Ginzburg-Landau equation. Using a combination of rigourous results and numerical computations, we describe a countable family of self-similar singularities. Our analysis includes the super-critical non-linear Schrödinger equation as a special case, and most of the described singularities are new even in that situation. We also consider the problem of stability of these singularities.