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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2402.14329 |
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| _version_ | 1866929252544282624 |
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| author | Papenburg, Hagen |
| author_facet | Papenburg, Hagen |
| contents | We prove unconditional local well-posedness in a space of quasi-periodic functions for dispersive equations of the form $$\partial_tu + Lu + \partial_x(u^{p+1})=0,$$ where $L$ is a multiplier operator with purely imaginary symbol which grows at most exponentially. The class of equations to which our method applies includes the generalized Korteweg-de Vries equation, the generalized Benjamin-Ono equation, and the derivative nonlinear Schrödinger equation. We also discuss well-posedness of some dispersive models which do not have a problematic derivative in the nonlinearity, namely, the nonlinear Schrödinger equation and the generalized Benjamin-Bona-Mahony equation, with quasi-periodic initial data. In this way, we recover and improve upon results from arXiv:1212.2674v3 [math.AP], arXiv:2110.11263v1 [math.AP] and arXiv:2201.02920v1 [math.AP] by shorter arguments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_14329 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Local Wellposedness of dispersive equations with quasi-periodic initial data Papenburg, Hagen Analysis of PDEs We prove unconditional local well-posedness in a space of quasi-periodic functions for dispersive equations of the form $$\partial_tu + Lu + \partial_x(u^{p+1})=0,$$ where $L$ is a multiplier operator with purely imaginary symbol which grows at most exponentially. The class of equations to which our method applies includes the generalized Korteweg-de Vries equation, the generalized Benjamin-Ono equation, and the derivative nonlinear Schrödinger equation. We also discuss well-posedness of some dispersive models which do not have a problematic derivative in the nonlinearity, namely, the nonlinear Schrödinger equation and the generalized Benjamin-Bona-Mahony equation, with quasi-periodic initial data. In this way, we recover and improve upon results from arXiv:1212.2674v3 [math.AP], arXiv:2110.11263v1 [math.AP] and arXiv:2201.02920v1 [math.AP] by shorter arguments. |
| title | Local Wellposedness of dispersive equations with quasi-periodic initial data |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2402.14329 |