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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Instability of algebraic standing waves for nonlinear Schrödinger equations with double power nonlinearities


Authors: Noriyoshi Fukaya and Masayuki Hayashi
Journal: Trans. Amer. Math. Soc. 374 (2021), 1421-1447
MSC (2010): Primary 35Q55; Secondary 35A15
DOI: https://doi.org/10.1090/tran/8269
Published electronically: November 25, 2020
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Abstract: We consider a nonlinear Schrödinger equation with double power nonlinearity

$\displaystyle i\partial _t u+\Delta u-\vert u\vert^{p-1}u+\vert u\vert^{q-1}u=0,\quad (t,x)\in \mathbb{R}\times \mathbb{R}^N,$    

where $ 1<p<q<1+4/(N-2)_+$. Due to the defocusing effect from the lower power order nonlinearity, the equation has algebraically decaying standing waves with zero frequency, which we call algebraic standing waves, as well as usual standing waves decaying exponentially with positive frequency. In this paper we study stability properties of two types of standing waves. We prove strong instability for all frequencies when $ q\ge 1+4/N$ and instability for small frequencies when $ q<1+4/N$, which especially give the first results on stability properties of algebraic standing waves. The instability result for small positive frequency when $ q<1+4/N$ not only improves previous results in the one-dimensional case but also gives a first result on instability in the higher-dimensional case. The key point in our approach is to take advantage of algebraic standing waves.

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Additional Information

Noriyoshi Fukaya
Affiliation: Department of Mathematics, Tokyo University of Science, Tokyo, 162-8601, Japan
Email: fukaya@rs.tus.ac.jp

Masayuki Hayashi
Affiliation: Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan
Email: hayashi@kurims.kyoto-u.ac.jp

DOI: https://doi.org/10.1090/tran/8269
Keywords: Nonlinear Schr\"odinger equation, standing waves, orbital instability, variational methods
Received by editor(s): February 11, 2020
Received by editor(s) in revised form: July 3, 2020
Published electronically: November 25, 2020
Additional Notes: The first author was supported by JSPS KAKENHI Grant Number 20K14349.
The second author was supported by JSPS KAKENHI Grant Number JP19J01504.
Article copyright: © Copyright 2020 American Mathematical Society