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Bifurcating bright and dark solitary waves for the perturbed cubic-quintic nonlinear Schrödinger equation

Published online by Cambridge University Press:  14 November 2011

Todd Kapitula
Todd Kapitula
Affiliation:
Department of Mathematics, Virginia Tech, Blacksburg, VA 24061-0123, U.S.A., E-mail: [email protected]
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Abstract

The existence of bright and dark multi-bump solitary waves for Ginzburg–Landau type perturbations of the cubic-quintic Schrodinger equation is considered. The waves in question are not perturbations of known analytic solitary waves, but instead arise as a bifurcation from a heteroclinic cycle in a three-dimensional ODE phase space. Using geometric singular perturbation techniques, regions in parameter space for which 1-bump bright and dark solitary waves will bifurcate are identified. The existence of N-bump dark solitary waves (N ≧ 1) is shown via an application of the Exchange Lemma with Exponentially Small Error. N-bump bright solitary waves are shown to exist as a consequence of the work of Kapitula and Maier-Paape.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 128 , Issue 3 , 1998 , pp. 585 - 629
Copyright
Copyright © Royal Society of Edinburgh 1998

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