标题： KdV 方程在非交换环面上的雅可比函数 显示英文标题

标题： KdV Equation for Theta Functions on Non-commutative Tori

摘要： 在非交换几何和弦理论领域，量子环面出现在不同的数学和物理情境中。因此，定义在量子环面上的量化θ函数也得到了研究（Yu. I. Manin, A. Schwartz；注意，两种量子θ函数定义之间的比较仍然是一个开放问题）。经典θ函数的一个重要应用是在孤立子理论中。某些孤立子方程，包括KdV方程，具有由θ函数给出的代数几何解（我们参考F. Gesztesy和H. Holden），因此属于“可积层次”。虽然量化可积性是一个非常活跃且复杂的研究领域，但在本工作中，我们采取了不同的、朴素的方法。我们进行一个实验：使用量子环面上的微分定义（M. Rieffel），我们询问量子θ函数是否满足非线性偏微分方程。该实验在2环面和KdV方程上取得了成功。这为将来的研究开辟了道路，例如寻找量子θ函数满足的相容层次，以及完全可积性的一致定义。 显示英文摘要

摘要： In the fields of non-commutative geometry and string theory, quantum tori appear in different mathematical and physical contexts. Therefore, quantized theta functions defined on quantum tori are also studied (Yu. I. Manin, A. Schwartz; note that a comparison between the two definitions of quantum theta is still an open problem). One important application of classical theta functions is in soliton theory. Certain soliton equations, including the KdV equation, have algebro-geometric solutions that are given by theta functions (we refer to F. Gesztesy and H. Holden), and as such belong to an "integrable hierarchy." While quantized integrability is a very active and complicated subject, in this work we take a different, naive approach. We conduct an experiment: using a definition of differentiation on quantum tori (M. Rieffel), we ask whether the quantum theta function satisfies non-linear PDE. The experiment is successful on the 2-torus and for the KdV equation. This opens the way to future investigations, such as the quest for a compatible hierarchy satisfied by quantum theta, and a consistent definition of complete integrability.