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Well-Posedness for KdV-Type Equations with Second-Order Derivative Nonlinearities

  • Mingjuan Chen
  • Ting Chen
  • Zhong Wang
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Abstract

We study a class of complex-valued KdV-type equations with cubic second-order derivative nonlinearities and arbitrary complex coefficients. For sufficiently small initial data in L2, we prove local well-posedness in Hs(ℝ) for s≥3/4. The key ingredient is a family of dyadic resolution spaces Zk=Xk+Yk, designed to overcome the logarithmic divergence arising from high*low*low interactions in which both derivatives fall on the highest-frequency factor. For the completely integrable third-order Kaup--Newell flow, we establish global-in-time Hs(ℝ) bounds for 0≤ s<1 and global well-posedness in Hs(ℝ) for every s≥3/4, with no smallness assumption on the initial data.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026wellposedness,
  title = {{Well-Posedness for KdV-Type Equations with Second-Order Derivative Nonlinearities}},
  author = {Mingjuan Chen and Ting Chen and Zhong Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.28975},
  eprint = {2609.28975},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.28975},
}
RIS
TY  - UNPB
TI  - Well-Posedness for KdV-Type Equations with Second-Order Derivative Nonlinearities
AU  - Mingjuan Chen
AU  - Ting Chen
AU  - Zhong Wang
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.28975
UR  - https://arxiv.org/abs/2609.28975
AB  - We study a class of complex-valued KdV-type equations with cubic second-order derivative nonlinearities and arbitrary complex coefficients. For sufficiently small initial data in L^2, we prove local well-posedness in H^s(ℝ) for s≥3/4. The key ingredient is a family of dyadic resolution spaces Z_k=X_k+Y_k, designed to overcome the logarithmic divergence arising from high*low*low interactions in which both derivatives fall on the highest-frequency factor. For the completely integrable third-order Kaup--Newell flow, we establish global-in-time H^s(ℝ) bounds for 0≤ s<1 and global well-posedness in H^s(ℝ) for every s≥3/4, with no smallness assumption on the initial data.
ER  -