Global solution and asymptotic behavior for the kinetic derivative NLS on $\mathbb R$
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915441468768256 |
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| author | Kishimoto, Nobu Lee, Kiyeon |
| author_facet | Kishimoto, Nobu Lee, Kiyeon |
| contents | In this paper we investigate the global well-posedness and long-term behavior of solutions to the kinetic derivative nonlinear Schrödinger equation (KDNLS) on the real line. The equation incorporates both local cubic nonlinearities with derivative terms and a non-local term arising from the Hilbert transform, modeling interactions in plasma physics. We establish global existence for small initial data in the weighted Sobolev space $H^2 \cap H^{1,1}$ and optimal time decay effect. Using energy methods and a frequency-localized gauge transformation, we overcome the difficulties posed by the non-local nonlinearities and provide a rigorous analysis of the asymptotic behavior. Our results also describe modified scattering phenomena with a suitable phase modification, showing that the solutions exhibit a precise asymptotic profile as $t \to \infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_08647 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global solution and asymptotic behavior for the kinetic derivative NLS on $\mathbb R$ Kishimoto, Nobu Lee, Kiyeon Analysis of PDEs In this paper we investigate the global well-posedness and long-term behavior of solutions to the kinetic derivative nonlinear Schrödinger equation (KDNLS) on the real line. The equation incorporates both local cubic nonlinearities with derivative terms and a non-local term arising from the Hilbert transform, modeling interactions in plasma physics. We establish global existence for small initial data in the weighted Sobolev space $H^2 \cap H^{1,1}$ and optimal time decay effect. Using energy methods and a frequency-localized gauge transformation, we overcome the difficulties posed by the non-local nonlinearities and provide a rigorous analysis of the asymptotic behavior. Our results also describe modified scattering phenomena with a suitable phase modification, showing that the solutions exhibit a precise asymptotic profile as $t \to \infty$. |
| title | Global solution and asymptotic behavior for the kinetic derivative NLS on $\mathbb R$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2508.08647 |