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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2511.06432 |
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| _version_ | 1866915638773022720 |
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| author | Chen, Xi |
| author_facet | Chen, Xi |
| contents | In this paper, we study the long time behavior of solutions to the defocusing Calogero--Moser derivative nonlinear Schrödinger equation (CM-DNLS). Using the Gérard-type explicit formula, we prove the scattering result of solutions to this equation with initial data in $L_{+}^{2,α}(\mathbb{R}): = \{u \in L_{+}^2(\mathbb{R}): |x|^α u \in L^2(\mathbb{R})\}$ with some $α>0$. We also characterize the scattering term using the distorted Fourier transform associated with the Lax operator. Following our approach developed in this paper, we can also conclude the asymptotic bound-state/radiation decomposition for global solutions to the focusing (CM-DNLS) with initial data in $L_{+}^{2,α}(\mathbb{R})$ with some $α>0$. This is one of the first works that apply the Gérard-type explicit formula to study the long-time behavior of an integrable equation for a broad class of initial data, beyond the previously studied rational cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_06432 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Scattering of the defocusing Calogero--Moser derivative nonlinear Schrödinger equation Chen, Xi Analysis of PDEs In this paper, we study the long time behavior of solutions to the defocusing Calogero--Moser derivative nonlinear Schrödinger equation (CM-DNLS). Using the Gérard-type explicit formula, we prove the scattering result of solutions to this equation with initial data in $L_{+}^{2,α}(\mathbb{R}): = \{u \in L_{+}^2(\mathbb{R}): |x|^α u \in L^2(\mathbb{R})\}$ with some $α>0$. We also characterize the scattering term using the distorted Fourier transform associated with the Lax operator. Following our approach developed in this paper, we can also conclude the asymptotic bound-state/radiation decomposition for global solutions to the focusing (CM-DNLS) with initial data in $L_{+}^{2,α}(\mathbb{R})$ with some $α>0$. This is one of the first works that apply the Gérard-type explicit formula to study the long-time behavior of an integrable equation for a broad class of initial data, beyond the previously studied rational cases. |
| title | Scattering of the defocusing Calogero--Moser derivative nonlinear Schrödinger equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2511.06432 |