Lower regularity well-posedness for a higher-order Schrödinger equation with cubic nonlinearities on the half-line
Abstract
In this paper, we continue the study of Himonas and Yan\cite{himonas2024schrodinger,himonas2026higher,himonas2024higher} on the nonlinear Schrödinger equation with a dispersion of order 2m and cubic nonlinearities, iut+( −1 ) m+1∂ x2mu=Nk( u,u,u ), where, m≥1 being an integer, N0( u,u,u ) =uuu, N1( u,u,u ) =u̅uu, N2( u,u,u ) =u̅u̅u, N3( u,u,u ) =u̅u̅u̅, We derive trilinear estimates at lower regularity and thereby prove that the cNLS-2m on the half-line is well-posed at the optimal regularity s=−frm−12. This improves the previous result, s>−frm−12, and answers an open question left in \cite{himonas2026higher}. In addition, for k=0,2, we establish well-posedness at s=−frm−12 as well. Moreover, for k=3, the nonlinearity exhibits a stronger resonance relation, which implies well-posedness for s>−fr2m−13. Our derivation of the trilinear estimates relies on varieties of Strichartz estimates, which differs from the [k;Z]-multiplier norm method employed in \cite{himonas2026higher}.
Related
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BibTeX
@misc{xi2026lower,
title = {{Lower regularity well-posedness for a higher-order Schrödinger equation with cubic nonlinearities on the half-line}},
author = {Yuhao Xi and Shenghao Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02747},
eprint = {2610.02747},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02747},
}
RIS
TY - UNPB
TI - Lower regularity well-posedness for a higher-order Schrödinger equation with cubic nonlinearities on the half-line
AU - Yuhao Xi
AU - Shenghao Li
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.02747
UR - https://arxiv.org/abs/2610.02747
AB - In this paper, we continue the study of Himonas and Yan\cite{himonas2024schrodinger,himonas2026higher,himonas2024higher} on the nonlinear Schrödinger equation with a dispersion of order 2m and cubic nonlinearities, iu_t+( −1 ) ^m+1∂ _x^2mu=N_k( u,u,u ), where, m≥1 being an integer, N_0( u,u,u ) =uuu, N_1( u,u,u ) =u̅uu, N_2( u,u,u ) =u̅u̅u, N_3( u,u,u ) =u̅u̅u̅, We derive trilinear estimates at lower regularity and thereby prove that the cNLS-2m on the half-line is well-posed at the optimal regularity s=−frm−12. This improves the previous result, s>−frm−12, and answers an open question left in \cite{himonas2026higher}. In addition, for k=0,2, we establish well-posedness at s=−frm−12 as well. Moreover, for k=3, the nonlinearity exhibits a stronger resonance relation, which implies well-posedness for s>−fr2m−13. Our derivation of the trilinear estimates relies on varieties of Strichartz estimates, which differs from the [k;Z]-multiplier norm method employed in \cite{himonas2026higher}.
ER -