Instability of stationary solutions for double power nonlinear Schrödinger equations in one dimension
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866916631268032512 |
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| author | Fukaya, Noriyoshi Hayashi, Masayuki |
| author_facet | Fukaya, Noriyoshi Hayashi, Masayuki |
| contents | We consider a double power nonlinear Schrödinger equation which possesses the algebraically decaying stationary solution $ϕ_0$ as well as exponentially decaying standing waves $e^{iωt}ϕ_ω(x)$ with $ω>0$. It is well-known from the general theory that stability properties of standing waves are determined by the derivative of $ω\mapsto M(ω):=\frac{1}{2}\|ϕ_ω\|_{L^2}^2$; namely $e^{iωt}ϕ_ω$ with $ω>0$ is stable if $M'(ω)>0$ and unstable if $M'(ω)<0$. However, the stability/instability of stationary solutions is outside the general theory from the viewpoint of spectral properties of linearized operators. In this paper we prove the instability of the stationary solution $ϕ_0$ in one dimension under the condition $M'(0):=\lim_{ω\downarrow 0}M'(ω)\in[-\infty, 0)$. The key in the proof is the construction of the one-sided derivative of $ω\mapstoϕ_ω$ at $ω=0$, which is effectively used to construct the unstable direction of $ϕ_0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_14337 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Instability of stationary solutions for double power nonlinear Schrödinger equations in one dimension Fukaya, Noriyoshi Hayashi, Masayuki Analysis of PDEs We consider a double power nonlinear Schrödinger equation which possesses the algebraically decaying stationary solution $ϕ_0$ as well as exponentially decaying standing waves $e^{iωt}ϕ_ω(x)$ with $ω>0$. It is well-known from the general theory that stability properties of standing waves are determined by the derivative of $ω\mapsto M(ω):=\frac{1}{2}\|ϕ_ω\|_{L^2}^2$; namely $e^{iωt}ϕ_ω$ with $ω>0$ is stable if $M'(ω)>0$ and unstable if $M'(ω)<0$. However, the stability/instability of stationary solutions is outside the general theory from the viewpoint of spectral properties of linearized operators. In this paper we prove the instability of the stationary solution $ϕ_0$ in one dimension under the condition $M'(0):=\lim_{ω\downarrow 0}M'(ω)\in[-\infty, 0)$. The key in the proof is the construction of the one-sided derivative of $ω\mapstoϕ_ω$ at $ω=0$, which is effectively used to construct the unstable direction of $ϕ_0$. |
| title | Instability of stationary solutions for double power nonlinear Schrödinger equations in one dimension |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2304.14337 |