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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Multiscale-bump standing waves with a critical frequency for nonlinear Schrödinger equations


Authors: Daomin Cao, Ezzat S. Noussair and Shusen Yan
Journal: Trans. Amer. Math. Soc. 360 (2008), 3813-3837
MSC (2000): Primary 35J20, 35J65
Published electronically: February 13, 2008
MathSciNet review: 2386247
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Abstract: In this paper we study the existence and qualitative property of standing wave solutions $ \psi(x,t) = e^{-\frac{iEt}{\hbar}} u(x)$ for the nonlinear Schrödinger equation $ i\hbar\frac{\partial\psi}{\partial t} + \frac{\hbar^2}{2m} \Delta \psi - W(x) \psi + \vert\psi\vert^{p-1} \psi = 0$ with $ E$ being a critical frequency in the sense that $ \inf\limits_{x\in \mathbb{R}^N} W(x)=E.$ We show that if the zero set of $ V=W-E$ has $ k$ isolated connected components $ Z_i (i=1,\cdots, k)$ such that the interior of $ Z_i$ is not empty and $ \partial Z_i$ is smooth, $ V$ has $ t$ isolated zero points, $ b_i$, $ i=1,\cdots,t$, and $ V$ has $ l$ critical points $ a_i(i=1,\cdots,l)$ such that $ V(a_i)>0$, then for $ \hbar > 0$ small, there exists a standing wave solution which is trapped in a neighborhood of $ \bigcup_{i=1} Z_i\cup\bigl(\bigcup_{i=1}^t\{b_i\})\cup \bigl(\bigcup_{i=1}^l\{a_i\}\bigr).$ Moreover the amplitudes of the standing wave around $ \bigcup^k_{i=1} Z_i$, $ \bigcup^t_{i=1}\{b_i\}$ and $ \bigcup^l_{i=1}\{a_i\}$ are of a different order of $ \hbar$. This type of multi-scale solution has never before been obtained.


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Additional Information

Daomin Cao
Affiliation: Institute of Applied Mathematics, AMSS, Chinese Academy of Sciences, Beijing 100080, People’s Republic of China
Email: dmcao@amt.ac.cn

Ezzat S. Noussair
Affiliation: School of Mathematics, The University of New South Wales, Sydney 2052, Australia
Email: noussair@maths.unsw.edu.au

Shusen Yan
Affiliation: School of Mathematics, Statistics and Computer Science, The University of New England, Armidale NSW 2351, Australia
Email: syan@turing.une.edu.au

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04348-1
PII: S 0002-9947(08)04348-1
Keywords: Multiscale-bump, standing waves, nonlinear Schr\"odinger equation, variational method, critical point
Received by editor(s): May 8, 2006
Received by editor(s) in revised form: June 15, 2006
Published electronically: February 13, 2008
Additional Notes: The first author was supported by the Fund of Distinguished Young Scholars of China and Innovative Funds of CAS in China
The second and third authors were supported by ARC in Australia
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.