The initial-boundary value problem for the nonlinear Schrödinger equation on the half-line with initial data in Sobolev spaces \(H^s(0, \infty )\), \(1/2< s\leqslant 5/2\), \(s\ne 3/2\), and Robin boundary data of appropriate regularity is shown to be locally well-posed in the sense of Hadamard. The proof is through a contraction mapping argument and hence relies crucially on certain estimates for the forced linear counterpart of the nonlinear problem. In particular, the essence of the analysis lies in the pure linear initial-boundary value problem, which corresponds to the case of zero forcing, zero initial data, and nonzero boundary data. This problem, which is studied by taking advantage of the solution formula derived via the unified transform of Fokas, holds an instrumental role in the overall analysis as it reveals the correct function space for the Robin boundary data.

非线性薛定谔方程的初边值问题在 Sobolev 空间\(H^s(0, \infty )\) , \(1/2< s\leqslant 5/2\) , \(s\ne 3/2\)，并且适当规律的罗宾边界数据在哈达玛的意义上被证明是局部适定的。证明是通过收缩映射论证，因此关键依赖于对非线性问题的强制线性对应物的某些估计。特别地，分析的本质在于纯线性初边值问题，它对应于迫零、零初始数据和非零边界数据的情况。该问题利用通过 Fokas 统一变换导出的解公式进行研究，在整体分析中发挥重要作用，因为它揭示了 Robin 边界数据的正确函数空间。