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- W1504236114 abstract "This paper is concerned with initial-boundary-value problems (IBVPs) for a class of nonlinear Schrodinger equations posed either on a half line $mathbb{R}^+$ or on a bounded interval $(0, L)$ with nonhomogeneous boundary conditions. For any $s$ with $0leq s < 5/2$ and $s not = 3/2$, it is shown that the relevant IBVPs are locally well-posed if the initial data lie in the $L^2$--based Sobolev spaces $H^s(mathbb{R}^+) $ in the case of the half line and in $H^s (0, L)$ on a bounded interval, provided the boundary data are selected from $H^{(2s+1)/4}_{loc} (mathbb{R}^+)$ and $H^{(s+ 1) /2}_{loc} (mathbb{R}^+)$, respectively. (For $s > frac12$, compatibility between the initial and boundary conditions is also needed.) Global well-posedness is also discussed when $s ge 1$. From the point of view of the well-posedness theory, the results obtained reveal a significant difference between the IBVP posed on $mathbb{R}^+$ and the IBVP posed on $(0,L)$. The former is reminiscent of the theory for the pure initial-value problem (IVP) for these Schrodinger equations posed on the whole line $mathbb{R}$ while the theory on a bounded interval looks more like that othe pure IVP posed on a periodic domain. In particular, the regularity demanded of the boundary data for the IBVP on $mathbb{R}^+$ is consistent with the temporal trace results that obtain for solutions of the pure IVP on $mathbb{R}$, while the slightly higher regularity of boundary data for the IBVP on $(0, L)$ resembles what is found for temporal traces of spatially periodic solutions." @default.
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- W1504236114 date "2015-02-27" @default.
- W1504236114 modified "2023-09-26" @default.
- W1504236114 title "Nonhomogeneous Boundary-Value Problems for One-Dimensional Nonlinear Schrodinger Equations" @default.
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- W1504236114 doi "https://doi.org/10.48550/arxiv.1503.00065" @default.
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