On the well-posedness of the periodic fractional Schrödinger equation
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| Format: | Preprint |
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2024
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| _version_ | 1866913540398383104 |
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| author | Sanchez, Beckett Riaño, Oscar Roudenko, Svetlana |
| author_facet | Sanchez, Beckett Riaño, Oscar Roudenko, Svetlana |
| contents | We consider the periodic fractional nonlinear Schrödinger equation $$ iu_t -(-Δ)^{\frac{s}{2}} u + \mathcal{N}(|u|)u=0, \quad x\in \mathbb{T}^N,\, \, t \in \mathbb R, \, \, s>0, $$ where the nonlinearity term is expressed in two ways: the first one $\mathcal{N}\in C^J(\mathbb R^+)$, whose derivatives have a certain polynomial decay, e.g., $\mathcal{N}(|u|)=\log(|u|)$; the second one is given by a sum of powers, possibly infinite, $$ \mathcal{N}(|u|) = \sum a_k |u|^{γ_k}, \quad γ_k \in \mathbb{R}, ~~ a_k \in \mathbb{C}, $$ which includes examples such as $\mathcal{N}(|u|) \, u =\frac{u}{|u|^γ},$ $γ>0$. By using standard properties of periodic Sobolev spaces $H^J(\mathbb{T}^N)$, $J>0$, we study the local well-posedness for the Cauchy problems of the above equations when initial data satisfy a non-vanishing condition $\inf\limits_{x\in \mathbb{T}^N}|u_0(x)|>0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_08144 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the well-posedness of the periodic fractional Schrödinger equation Sanchez, Beckett Riaño, Oscar Roudenko, Svetlana Analysis of PDEs 35A01, 35B10, 35Q40, 35Q55 We consider the periodic fractional nonlinear Schrödinger equation $$ iu_t -(-Δ)^{\frac{s}{2}} u + \mathcal{N}(|u|)u=0, \quad x\in \mathbb{T}^N,\, \, t \in \mathbb R, \, \, s>0, $$ where the nonlinearity term is expressed in two ways: the first one $\mathcal{N}\in C^J(\mathbb R^+)$, whose derivatives have a certain polynomial decay, e.g., $\mathcal{N}(|u|)=\log(|u|)$; the second one is given by a sum of powers, possibly infinite, $$ \mathcal{N}(|u|) = \sum a_k |u|^{γ_k}, \quad γ_k \in \mathbb{R}, ~~ a_k \in \mathbb{C}, $$ which includes examples such as $\mathcal{N}(|u|) \, u =\frac{u}{|u|^γ},$ $γ>0$. By using standard properties of periodic Sobolev spaces $H^J(\mathbb{T}^N)$, $J>0$, we study the local well-posedness for the Cauchy problems of the above equations when initial data satisfy a non-vanishing condition $\inf\limits_{x\in \mathbb{T}^N}|u_0(x)|>0$. |
| title | On the well-posedness of the periodic fractional Schrödinger equation |
| topic | Analysis of PDEs 35A01, 35B10, 35Q40, 35Q55 |
| url | https://arxiv.org/abs/2410.08144 |