Year: 2026
Type: preprint
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 $L^2$, we prove local well-posedness in $H^s(\mathbb R)$ for $s\ge3/4$. 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 $H^s(\mathbb R)$ bounds for $0\le s<1$ and global well-posedness in $H^s(\mathbb R)$ for every $s\ge3/4$, with no smallness assumption on the initial data.
Source: arXiv (Cornell University)
Authors: Chen, Mingjuan, Chen, Ting, Wang, Zhong
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G1968463374, G2180011517, G5666607446
OpenAlex ID: W7214342247