Lower regularity well-posedness for a higher-order Schrödinger equation with cubic nonlinearities on the half-line
Year: 2026
Type: preprint
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, \[ iu_t+\left( -1 \right) ^{m+1}\partial _{x}^{2m}u=N_k\left( u,u,u \right), \] where, $m\ge1$ being an integer, \[N_0\left( u,u,u \right) =uuu,\quad N_1\left( u,u,u \right) =\bar{u}uu,\quad N_2\left( u,u,u \right) =\bar{u}\bar{u}u,\quad N_3\left( u,u,u \right) =\bar{u}\bar{u}\bar{u},\] 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 $s=-\fr{m-1}{2}$. This improves the previous result, $s>-\fr{m-1}{2}$, and answers an open question left in \cite{himonas2026higher}. In addition, for $k=0,2$, we establish well-posedness at $s=-\fr{m-1}{2}$ as well. Moreover, for $k=3$, the nonlinearity exhibits a stronger resonance relation, which implies well-posedness for $s>-\fr{2m-1}{3}$. 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}.
Source: arXiv (Cornell University)
Authors: Xi, Yuhao, Li, Shenghao
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G869107361
OpenAlex ID: W7220378480