Author

Rui-Xiao Zhang

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Preprint Oct 2026

The quadratic Schr\"odinger system on $\mathbb{T}^6$

In this article, we study the global well-posedness for the quadratic Schr\"odinger system on $\mathbb{T}^6$. To achieve this, we utilize the concentration-compactness/rigitdity argument initiated by Ionescu and Pausader \cite{IPT3}. Our result is also regarded as a periodic analogue of Gao-Meng-Xu-Zheng \cite{GMXZ}.

Yi-Lin Song, Yan Xue, Rui-Xiao Zhang · 0 citations

Preprint Oct 2026

A sharp isoperimetric bound for the ratio of the fifth and fourth eigenvalues of Dirichlet Laplacian

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with $n\geq2$. The well-known Payne-P\'olya-Weinberger conjecture for consecutive eigenvalue ratios asserts that $$ \frac{\lambda_{m+1}(\Omega)}{\lambda_m(\Omega)}\leq\left(\frac{j_{n / 2,1}}{j_{n / 2-1,1}}\right)^2,\quad m=1,2,3,\cdots. $$ Here, $j_{p,k}$ denotes the...

Zhen-Guo Song, Rui-Xiao Zhang · 0 citations