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Infinitely many radial solutions of a variational problem related to dispersion-managed optical fibers


Author: Markus Kunze
Journal: Proc. Amer. Math. Soc. 131 (2003), 2181-2188
MSC (1991): Primary 35A15, 35Q55; Secondary 78A60
DOI: https://doi.org/10.1090/S0002-9939-02-06780-1
Published electronically: November 13, 2002
MathSciNet review: 1963766
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Abstract: We consider a non-local variational problem whose critical points are related to bound states in certain optical fibers. The functional is given by $\varphi(u)=\frac{1}{2}\vert u\vert^2_{H^1}-\int_0^1\int_{\mathbb R^2}\vert e^{it\Delta}u\vert^4\,dxdt$, and relying on the regularizing properties of the solution $e^{it\Delta}$ to the free Schrödinger equation, it will be shown that $\varphi$ has infinitely many critical points.


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Additional Information

Markus Kunze
Affiliation: FB 6 – Mathematik, Universität Essen, D-45117 Essen, Germany
Email: mkunze@ing-math.uni-essen.de

DOI: https://doi.org/10.1090/S0002-9939-02-06780-1
Keywords: Nonlocal variational problem, compactness by symmetry, infinitely many solutions, nonlinear optics, dispersion managed solitons
Received by editor(s): December 13, 2001
Received by editor(s) in revised form: March 3, 2002
Published electronically: November 13, 2002
Communicated by: Andreas Seeger
Article copyright: © Copyright 2002 American Mathematical Society

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