Long-range scattering for a critical homogeneous type nonlinear Schrödinger equation with time-decaying harmonic potentials
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866929265174380544 |
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| author | Kawamoto, Masaki Miyazaki, Hayato |
| author_facet | Kawamoto, Masaki Miyazaki, Hayato |
| contents | This paper is concerned with the final state problem for the homogeneous type nonlinear Schrödinger equation with time-decaying harmonic potential. The nonlinearity has the critical order and is not necessarily the form of a polynomial. In the case of the gauge-invariant power-type nonlinearity, the first author proves that the equation admits a nontrivial solution that behaves like a free solution with a logarithmic phase correction in [22]. In this paper, we extend his result into the case with the general homogeneous nonlinearity by the technique due to the Fourier series expansion introduced by Masaki and the second author [26]. To adapt the argument in the aforementioned paper, we develop a factorization identity for the propagator and require a little stronger decay condition for the Fourier coefficients arising from the harmonic potential. Moreover, in two or three dimensions, we improve the regularity condition of the final data in [26, 28]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_08168 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Long-range scattering for a critical homogeneous type nonlinear Schrödinger equation with time-decaying harmonic potentials Kawamoto, Masaki Miyazaki, Hayato Analysis of PDEs 35Q55 (Primary) 35B40, 35P25 (Secondly) This paper is concerned with the final state problem for the homogeneous type nonlinear Schrödinger equation with time-decaying harmonic potential. The nonlinearity has the critical order and is not necessarily the form of a polynomial. In the case of the gauge-invariant power-type nonlinearity, the first author proves that the equation admits a nontrivial solution that behaves like a free solution with a logarithmic phase correction in [22]. In this paper, we extend his result into the case with the general homogeneous nonlinearity by the technique due to the Fourier series expansion introduced by Masaki and the second author [26]. To adapt the argument in the aforementioned paper, we develop a factorization identity for the propagator and require a little stronger decay condition for the Fourier coefficients arising from the harmonic potential. Moreover, in two or three dimensions, we improve the regularity condition of the final data in [26, 28]. |
| title | Long-range scattering for a critical homogeneous type nonlinear Schrödinger equation with time-decaying harmonic potentials |
| topic | Analysis of PDEs 35Q55 (Primary) 35B40, 35P25 (Secondly) |
| url | https://arxiv.org/abs/2206.08168 |