Time splitting method for nonlinear Schrödinger equation with rough initial data in $L^2$
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913580179259392 |
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| author | Choi, Hyung Jun Kim, Seonghak Koh, Youngwoo |
| author_facet | Choi, Hyung Jun Kim, Seonghak Koh, Youngwoo |
| contents | We establish convergence results related to the operator splitting scheme on the Cauchy problem for the nonlinear Schrödinger equation with rough initial data in $L^2$,
$$
\left\{
\begin{array}{ll}
i\partial_t u +Δu = λ|u|^{p} u, & (x,t) \in \mathbb{R}^d \times \mathbb{R}_+,
u (x,0) =ϕ(x), & x\in\mathbb{R}^d,
\end{array}
\right.
$$ where $λ\in \{-1,1\}$ and $p >0$. While the Lie approximation $Z_L$ is known to converge to the solution $u$ when the initial datum $ϕ$ is sufficiently smooth, the convergence result for rough initial data is open to question. In this paper, for rough initial data $ϕ\in L^2 (\mathbb{R}^d)$, we prove the $L^2$ convergence of the filtered Lie approximation $Z_{flt}$ to the solution $u$ in the mass-subcritical range, $0< p < \frac{4}{d}$. Furthermore, we provide a precise convergence result for radial initial data $ϕ\in L^2 (\mathbb{R}^d)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_07410 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Time splitting method for nonlinear Schrödinger equation with rough initial data in $L^2$ Choi, Hyung Jun Kim, Seonghak Koh, Youngwoo Numerical Analysis Analysis of PDEs 35Q55, 65M15 We establish convergence results related to the operator splitting scheme on the Cauchy problem for the nonlinear Schrödinger equation with rough initial data in $L^2$, $$ \left\{ \begin{array}{ll} i\partial_t u +Δu = λ|u|^{p} u, & (x,t) \in \mathbb{R}^d \times \mathbb{R}_+, u (x,0) =ϕ(x), & x\in\mathbb{R}^d, \end{array} \right. $$ where $λ\in \{-1,1\}$ and $p >0$. While the Lie approximation $Z_L$ is known to converge to the solution $u$ when the initial datum $ϕ$ is sufficiently smooth, the convergence result for rough initial data is open to question. In this paper, for rough initial data $ϕ\in L^2 (\mathbb{R}^d)$, we prove the $L^2$ convergence of the filtered Lie approximation $Z_{flt}$ to the solution $u$ in the mass-subcritical range, $0< p < \frac{4}{d}$. Furthermore, we provide a precise convergence result for radial initial data $ϕ\in L^2 (\mathbb{R}^d)$. |
| title | Time splitting method for nonlinear Schrödinger equation with rough initial data in $L^2$ |
| topic | Numerical Analysis Analysis of PDEs 35Q55, 65M15 |
| url | https://arxiv.org/abs/2305.07410 |