Local well-posedness for the derivative nonlinear Schrödinger equation with nonvanishing boundary conditions
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915307194417152 |
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| author | Molinet, Luc Tanaka, Tomoyuki |
| author_facet | Molinet, Luc Tanaka, Tomoyuki |
| contents | We consider the derivative nonlinear Schrödinger equation on the real line, with a background function $ψ(t,x)\in L^\infty(\mathbb{R}^2)$ that satisfies suitable conditions. Such a function may, for example, be a non-decaying solution of the equation, such as a dark soliton. By developing the energy method with correction terms, we prove that the Cauchy problem for perturbations around such an $L^\infty$ function is unconditionally locally well-posed in $ H^s(\mathbb{R}) $ for $ s>3/4 $. As a byproduct, we also establish local well-posedness in the Zhidkov space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_20883 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Local well-posedness for the derivative nonlinear Schrödinger equation with nonvanishing boundary conditions Molinet, Luc Tanaka, Tomoyuki Analysis of PDEs We consider the derivative nonlinear Schrödinger equation on the real line, with a background function $ψ(t,x)\in L^\infty(\mathbb{R}^2)$ that satisfies suitable conditions. Such a function may, for example, be a non-decaying solution of the equation, such as a dark soliton. By developing the energy method with correction terms, we prove that the Cauchy problem for perturbations around such an $L^\infty$ function is unconditionally locally well-posed in $ H^s(\mathbb{R}) $ for $ s>3/4 $. As a byproduct, we also establish local well-posedness in the Zhidkov space. |
| title | Local well-posedness for the derivative nonlinear Schrödinger equation with nonvanishing boundary conditions |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2505.20883 |