Local well-posedness of the higher-order nonlinear Schrödinger equation on the half-line: the case of two boundary conditions

@misc{alkın2026local,
  title = {{Local well-posedness of the higher-order nonlinear Schrödinger equation on the half-line: the case of two boundary conditions}},
  author = {Aykut Alkın and Dionyssios Mantzavinos and Türker Özsarı},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.03558},
  eprint = {2609.03558},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.03558},
}

TY  - UNPB
TI  - Local well-posedness of the higher-order nonlinear Schrödinger equation on the half-line: the case of two boundary conditions
AU  - Aykut Alkın
AU  - Dionyssios Mantzavinos
AU  - Türker Özsarı
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2609.03558
UR  - https://arxiv.org/abs/2609.03558
AB  - We prove the local Hadamard well-posedness of the higher-order nonlinear Schrödinger equation with negative dispersion coefficient on the half-line under nonzero Dirichlet and Neumann boundary data. The paper uncovers a striking feature of this model: unlike the positive-dispersion case studied in (Alkın, Mantzavinos, Ozsari, 2024), well-posedness requires two boundary conditions rather than one. This reflects a structural limitation in the underlying global relation, which allows elimination of at most one unknown boundary trace. Using the unified transform of Fokas, we derive an explicit solution formula for the associated forced linear problem and establish sharp Sobolev and Strichartz estimates on the half-line. These linear estimates, together with corresponding Cauchy problem bounds, yield local well-posedness in both high- and low-regularity Sobolev spaces, as well as local Lipschitz continuity of the solution map.
ER  -