Traveling Waves for Nonlinear Schrödinger Equations with Nonzero Conditions at Infinity

Traveling Waves for Nonlinear Schrödinger Equations with Nonzero Conditions at Infinity We prove the existence of nontrivial finite energy traveling waves for a large class of nonlinear Schrödinger equations with nonzero conditions at infinity (includindg the Gross–Pitaevskii and the so-called “cubic-quintic” equations) in space dimension $${ N \geq 2}$$ N ≥ 2 . We show that minimization of the energy at fixed momentum can be used whenever the associated nonlinear potential is nonnegative and it gives a set of orbitally stable traveling waves, while minimization of the action at constant kinetic energy can be used in all cases. We also explore the relationship between the families of traveling waves obtained by different methods and we prove a sharp nonexistence result for traveling waves with small energy. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archive for Rational Mechanics and Analysis Springer Journals

Traveling Waves for Nonlinear Schrödinger Equations with Nonzero Conditions at Infinity

100 pages
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Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2017 by Springer-Verlag Berlin Heidelberg
Subject
Physics; Classical Mechanics; Physics, general; Theoretical, Mathematical and Computational Physics; Complex Systems; Fluid- and Aerodynamics
ISSN
0003-9527
eISSN
1432-0673
D.O.I.
10.1007/s00205-017-1131-2
Publisher site
See Article on Publisher Site

Abstract

We prove the existence of nontrivial finite energy traveling waves for a large class of nonlinear Schrödinger equations with nonzero conditions at infinity (includindg the Gross–Pitaevskii and the so-called “cubic-quintic” equations) in space dimension $${ N \geq 2}$$ N ≥ 2 . We show that minimization of the energy at fixed momentum can be used whenever the associated nonlinear potential is nonnegative and it gives a set of orbitally stable traveling waves, while minimization of the action at constant kinetic energy can be used in all cases. We also explore the relationship between the families of traveling waves obtained by different methods and we prove a sharp nonexistence result for traveling waves with small energy.

Journal

Archive for Rational Mechanics and AnalysisSpringer Journals

Published: Jun 19, 2017

References

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