Differential Equations characterising quasi-periodic Solutions of the KP hierarchy

利用代数集刻画半 Clifford 门

Characterising semi-Clifford gates using algebraic sets

本文研究容错量子计算中的 Clifford 层级门，证明任意素数维度下最多两个 qudit 的第三层门均为半 Clifford 门，并利用代数几何构造概形，证明其模 $d$ 约化后的代数集有理点相同，推广了早期结果。

On quasi-periodic solutions of the discrete Chen-Lee-Liu hierarchy

Resorting to the Lax matrix and elliptic variables, the discrete Chen-Lee-Liu hierarchy is decomposed into solvable ordinary differential equations. Based on the theory of algebraic curve, the continuous flow and discrete flow related to the discrete Chen-Lee-Liu hierarchy are straightened under the Abel-Jacobi coordinates. The meromorphic function $ϕ$, the Baker-Akhiezer vector $\barψ$ and the hyperelliptic curve $\mathcal{K}_N$ are introduced, by which quasi-periodic solutions of the discrete Chen-Lee-Liu hierarchy are constructed according to the asymptotic properties and the algebro-geomet…

KP 族的顶点算子与奇异代数曲线

Vertex Operators of the KP hierarchy and Singular Algebraic Curves

本文利用 Sato Grassmannian 证明 KP 族顶点算子作用产生的解对应奇异代数曲线上的无挠秩一 sheaf，并说明其可视为拟周期背景上的孤子。

MKdV方程的精确准周期解

Exact quasi-periodic solutions to the MKdV equation

本研究利用代数几何方法求解mKdV方程层级，基于超椭圆谱曲线与推广的$\u200b\wp$-函数得到任意亏格的有限间隙通解，明确准周期波解实性条件，并以小亏格绘图验证，为非线性方程精确解研究提供了代数几何范式。

KdV方程的现实条件与有限相空间中的精确准周期解

Reality conditions for the KdV equation and exact quasi-periodic solutions in finite phase spaces

本文完整确定了KdV方程准周期解的现实条件，证明任意亏格超椭圆曲线雅可比簇上的多周期$\wp_{1,1}$函数即为有限间隙解，并给出有效计算方法。