Bifurcation revisited along footprints of Jürgen Scheurle

Tassilo Küpper

doi:10.3934/dcdss.2020061

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2020, Volume 13, Issue 4: 1031-1041. Doi: 10.3934/dcdss.2020061

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Bifurcation revisited along footprints of Jürgen Scheurle

Tassilo Küpper

Mathematisches Institut, Universität zu Köln, Weyertal 86-90, D50931 Köln, Germany

Received: August 31, 2017

Revised: May 31, 2018

Published: April 2020

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Abstract

Actual research concerning, in particular, the occurrence of "gap-solitons" bifurcating from the continuous spectrum confirms that this part of Bifurcation Theory that started around 40 years ago flourishes. In this lecture we review the origins of "Bifurcation from the continuous spectrum" with regard to the achievements of Jürgen Scheurle and sketch how the early results dealing with the bifurcation of singular solutions have prepared the ground for present and further developments.

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Mathematics Subject Classification: Primary: 34C23, 37G99; Secondary: 35J20, 47E05.

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Figure 1. Phase portrait for $ \lambda >0 $

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Figure 2. Heteroclinic orbit (singular solution)

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Figure 3. (a) λ > 0, (b) λ < 0

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Figure 4. Prof. Tassilo Küpper and Prof. Jürgen Scheurle

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