| Abstract: |
| This work studies the initial-boundary value problem on the half-line for the cubic nonlinear Schr\odinger equation with a dispersion of order $m=2, 4, \cdots$. The main result obtained is the optimal well-posedness of this problem when the initial data belong in the spatial Sobolev spaces $H^s(0,\infty)$, $s>-\frac14m +\frac12$, and the boundary data belong in appropriate temporal Sobolev spaces suggested by the time regularity of the linear problem. The proof is based on the Fokas solution formula for the forced linear problem and the linear estimates obtained for this solution in Bourgain spaces. Deriving sharp trilinear estimates suggested by the linear estimates and then applying them, it is shown that the iteration map defined by the Fokas solution formula is a contraction in an appropriate solution space. |
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