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PreprintCC BY 4.0arXiv

Well-posedness for the Cubic Nonlinear Schrödinger Equation With Nonlinearity Supported on a Hypersurface

  • Xilin Chen
  • Jianlei Huang
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Abstract

We prove that the 2-d cubic nonlinear Schrödinger equation with nonlinearity supported on a line or a sphere is globally well-posed in H1(ℝ2). For higher dimensions, we prove that the equation with hyperplane-supported nonlinearity is locally well-posed in suitable anisotropic Sobolev spaces. We also prove that the equation with sphere-supported nonlinearity is globally well-posed in H1(ℝn) for radial data.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026wellposedness,
  title = {{Well-posedness for the Cubic Nonlinear Schrödinger Equation With Nonlinearity Supported on a Hypersurface}},
  author = {Xilin Chen and Jianlei Huang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.32180},
  eprint = {2610.32180},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.32180},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Well-posedness for the Cubic Nonlinear Schrödinger Equation With Nonlinearity Supported on a Hypersurface
AU  - Xilin Chen
AU  - Jianlei Huang
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2610.32180
UR  - https://arxiv.org/abs/2610.32180
AB  - We prove that the 2-d cubic nonlinear Schrödinger equation with nonlinearity supported on a line or a sphere is globally well-posed in H^1(ℝ^2). For higher dimensions, we prove that the equation with hyperplane-supported nonlinearity is locally well-posed in suitable anisotropic Sobolev spaces. We also prove that the equation with sphere-supported nonlinearity is globally well-posed in H^1(ℝ^n) for radial data.
ER  -