Modified scattering operator for nonlinear Schrödinger equations with time-decaying harmonic potentials
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929264690987008 |
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| author | Kawamoto, Masaki Miyazaki, Hayato |
| author_facet | Kawamoto, Masaki Miyazaki, Hayato |
| contents | This paper is concerned with nonlinear Schrödinger equations with a time-decaying harmonic potential. The nonlinearity is gauge-invariant of the long-range critical order. In [24] and [22], it is proved that the equation admits a nontrivial solution that behaves like a free solution with a logarithmic phase correction in the frameworks of both the final state problem and the initial value problem. Furthermore, a modified scattering operator has been established in the case without the potential in [15]. In this paper, we construct a modified scattering operator for our equation by utilizing a generator of the Galilean transformation. Moreover, we remove a restriction for the coefficient of the potential which is required in [22]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_02657 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Modified scattering operator for nonlinear Schrödinger equations with time-decaying harmonic potentials Kawamoto, Masaki Miyazaki, Hayato Analysis of PDEs 35Q55, 35B40, 35P25 This paper is concerned with nonlinear Schrödinger equations with a time-decaying harmonic potential. The nonlinearity is gauge-invariant of the long-range critical order. In [24] and [22], it is proved that the equation admits a nontrivial solution that behaves like a free solution with a logarithmic phase correction in the frameworks of both the final state problem and the initial value problem. Furthermore, a modified scattering operator has been established in the case without the potential in [15]. In this paper, we construct a modified scattering operator for our equation by utilizing a generator of the Galilean transformation. Moreover, we remove a restriction for the coefficient of the potential which is required in [22]. |
| title | Modified scattering operator for nonlinear Schrödinger equations with time-decaying harmonic potentials |
| topic | Analysis of PDEs 35Q55, 35B40, 35P25 |
| url | https://arxiv.org/abs/2403.02657 |