Nonhomogeneous Boundary-Value Problems for One-Dimensional Nonlinear Schrödinger Equations

Abstract

This paper is concerned with initial-boundary-value problems (IBVPs) for a class of nonlinear Schrödinger equations posed either on a half line ${\mathbb{R}}^{+}$ or on a bounded interval $(0,L)$ with nonhomogeneous boundary conditions. For any $s$ with $0\le s<5/2$ and $s\ne 3/2$, 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 $H_{loc}^{(2s+1)/4}({\mathbb{R}}^{+})$ and $H_{loc}^{(s+1)/2}({\mathbb{R}}^{+})$, respectively. (For $s>\frac{1}{2}$, 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 Schrödinger 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.

This is a revised version

Cited by in corpus (2)

• Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity I: Basic theory
• Low-regularity global well-posedness for the Klein-Gordon-Schrödinger system on ${\mathbb{R}}^{+}$