Stability of standing waves for all frequencies to nonlinear Schrödinger equations with potentials in one dimension
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| Format: | Preprint |
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2025
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| _version_ | 1866914059099570176 |
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| author | Fukaya, Noriyoshi Ikeda, Masahiro Kikuchi, Hiroaki |
| author_facet | Fukaya, Noriyoshi Ikeda, Masahiro Kikuchi, Hiroaki |
| contents | In this paper, we study the orbital stability of standing waves for one-dimensional nonlinear Schrödinger equations with potentials. We show that the standing waves are orbitally stable for all frequencies in the $L^{2}$- subcritical and critical cases. Since the presence of potentials breaks the scale invariance of the equations, it is a delicate problem to apply the abstract theory of Grillakis, Shatah, and Strauss (1987) directly without a perturbative argument. For this reason, little is known about the orbital stability of standing waves for \textit{all} frequencies in the non-scale-invariant setting. We overcome this difficulty by employing the approach of Noris, Tavares, and Verzini (2014). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_23016 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stability of standing waves for all frequencies to nonlinear Schrödinger equations with potentials in one dimension Fukaya, Noriyoshi Ikeda, Masahiro Kikuchi, Hiroaki Analysis of PDEs 35Q55, 35B35 In this paper, we study the orbital stability of standing waves for one-dimensional nonlinear Schrödinger equations with potentials. We show that the standing waves are orbitally stable for all frequencies in the $L^{2}$- subcritical and critical cases. Since the presence of potentials breaks the scale invariance of the equations, it is a delicate problem to apply the abstract theory of Grillakis, Shatah, and Strauss (1987) directly without a perturbative argument. For this reason, little is known about the orbital stability of standing waves for \textit{all} frequencies in the non-scale-invariant setting. We overcome this difficulty by employing the approach of Noris, Tavares, and Verzini (2014). |
| title | Stability of standing waves for all frequencies to nonlinear Schrödinger equations with potentials in one dimension |
| topic | Analysis of PDEs 35Q55, 35B35 |
| url | https://arxiv.org/abs/2509.23016 |