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

Authors:	Daomin Cao Ezzat S Noussair Shusen Yan

Institution:	Institute of Applied Mathematics, AMSS, Chinese Academy of Sciences, Beijing 100080, People's Republic of China ; School of Mathematics, The University of New South Wales, Sydney 2052, Australia ; School of Mathematics, Statistics and Computer Science, The University of New England, Armidale NSW 2351, Australia

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 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 has isolated connected components $Z_i (i=1,\cdots, k)$ such that the interior of is not empty and $\partial Z_i$ is smooth, has isolated zero points, , $i=1,\cdots,t$ , and has critical points $a_i(i=1,\cdots,l)$ such that , 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.

Keywords:	Multiscale-bump standing waves nonlinear Schr\"odinger equation variational method critical point

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