$L^{2}$-Sobolev space bijectivity and existence of global solutions for the matrix nonlinear Schrödinger equations
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866914924930793472 |
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| author | Li, Yuan Liu, Xinhan Fan, Engui |
| author_facet | Li, Yuan Liu, Xinhan Fan, Engui |
| contents | We consider the Cauchy problem to the general defocusing and focusing $p\times q$ matrix nonlinear Schrödinger (NLS) equations with initial data allowing arbitrary-order poles and spectral singularities. By establishing the $L^{2}$-Sobolev space bijectivity of the direct and inverse scattering transforms associated with a $(p+q)\times(p+q)$ matrix spectral problem, we prove that both defocusing and focusing matrix NLS equations are globally well-posed in the weighted Sobolev space $H^{1,1}(\mathbb{R})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_14709 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $L^{2}$-Sobolev space bijectivity and existence of global solutions for the matrix nonlinear Schrödinger equations Li, Yuan Liu, Xinhan Fan, Engui Analysis of PDEs Mathematical Physics 35P25, 35Q15, 35Q35, 35A01 We consider the Cauchy problem to the general defocusing and focusing $p\times q$ matrix nonlinear Schrödinger (NLS) equations with initial data allowing arbitrary-order poles and spectral singularities. By establishing the $L^{2}$-Sobolev space bijectivity of the direct and inverse scattering transforms associated with a $(p+q)\times(p+q)$ matrix spectral problem, we prove that both defocusing and focusing matrix NLS equations are globally well-posed in the weighted Sobolev space $H^{1,1}(\mathbb{R})$. |
| title | $L^{2}$-Sobolev space bijectivity and existence of global solutions for the matrix nonlinear Schrödinger equations |
| topic | Analysis of PDEs Mathematical Physics 35P25, 35Q15, 35Q35, 35A01 |
| url | https://arxiv.org/abs/2408.14709 |