Lower regularity well-posedness for a higher-order Schrödinger equation with cubic nonlinearities on the half-line

Yuhao Xi; Shenghao Li

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arXiv · 2610.02747

Lower regularity well-posedness for a higher-order Schrödinger equation with cubic nonlinearities on the half-line

Yuhao Xi·Shenghao Li

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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}.

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BibTeX RIS

Yuhao Xi, Shenghao Li. 2026-10-02. Lower regularity well-posedness for a higher-order Schrödinger equation with cubic nonlinearities on the half-line. https://arxiv.org/abs/2610.02747

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