Instability of algebraic standing waves for nonlinear Schrödinger equations with double power nonlinearities
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866912245846376448 |
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| author | Fukaya, Noriyoshi Hayashi, Masayuki |
| author_facet | Fukaya, Noriyoshi Hayashi, Masayuki |
| contents | We consider a nonlinear Schrödinger equation with double power nonlinearity \begin{align*} i\partial_t u+Δu-|u|^{p-1}u+|u|^{q-1}u=0,\quad (t,x)\in\mathbb{R}\times\mathbb{R}^N, \end{align*} 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 one-dimensional case but also gives a first result on instability in higher-dimensional case. The key point in our approach is to take advantage of algebraic standing waves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2001_08488 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Instability of algebraic standing waves for nonlinear Schrödinger equations with double power nonlinearities Fukaya, Noriyoshi Hayashi, Masayuki Analysis of PDEs 35Q55, 35A15 We consider a nonlinear Schrödinger equation with double power nonlinearity \begin{align*} i\partial_t u+Δu-|u|^{p-1}u+|u|^{q-1}u=0,\quad (t,x)\in\mathbb{R}\times\mathbb{R}^N, \end{align*} 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 one-dimensional case but also gives a first result on instability in higher-dimensional case. The key point in our approach is to take advantage of algebraic standing waves. |
| title | Instability of algebraic standing waves for nonlinear Schrödinger equations with double power nonlinearities |
| topic | Analysis of PDEs 35Q55, 35A15 |
| url | https://arxiv.org/abs/2001.08488 |