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Local bifurcations of plane running waves for the generalized cubic Schrödinger equation

Local bifurcations of plane running waves for the generalized cubic Schrödinger equation For the generalized cubic Schrödinger equation, we consider a periodic boundary value problem in the case of n independent space variables. For this boundary value problem, there exists a countable set of plane running waves periodic with respect to the time variable. We analyze their stability and local bifurcations under the change of stability. We show that invariant tori of dimension 2, ..., n + 1 can bifurcate from each of them. We obtain asymptotic formulas for the solutions on invariant tori and stability conditions for bifurcating tori as well as parameter ranges in which, starting from n = 3, a subcritical (stiff) bifurcation of invariant tori is possible. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Differential Equations Springer Journals

Local bifurcations of plane running waves for the generalized cubic Schrödinger equation

Kulikov, A.; Kulikov, D.
Differential Equations , Volume 46 (9) – Nov 5, 2010
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References (17)

Publisher
Springer Journals
Copyright
Copyright © 2010 by Pleiades Publishing, Ltd.
Subject
Mathematics; Difference and Functional Equations; Partial Differential Equations; Ordinary Differential Equations
ISSN
0012-2661
eISSN
1608-3083
DOI
10.1134/S0012266110090065
Publisher site
See Article on Publisher Site

Abstract

For the generalized cubic Schrödinger equation, we consider a periodic boundary value problem in the case of n independent space variables. For this boundary value problem, there exists a countable set of plane running waves periodic with respect to the time variable. We analyze their stability and local bifurcations under the change of stability. We show that invariant tori of dimension 2, ..., n + 1 can bifurcate from each of them. We obtain asymptotic formulas for the solutions on invariant tori and stability conditions for bifurcating tori as well as parameter ranges in which, starting from n = 3, a subcritical (stiff) bifurcation of invariant tori is possible.

Journal

Differential Equations – Springer Journals

Published: Nov 5, 2010

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