arXiv 2402.180792024-02-28nlin.SImath-phmath.DSmath.MP

Neumann-Moser动力系统与Korteweg-de Vries层级

The Neumann-Moser dynamical system and the Korteweg-de Vries hierarchy

Polina Baron

AI总结：

本文应用Moser变换构造$\mathbb{C}^{3n+1}$上的二次可积Neumann-Moser系统，给出逆变换及积分对应，并证明其与Mumford系统和稳态KdV层级等价，进而用超椭圆Kleinian函数构造显式循环解。

摘要：

本文的重点是C. Neumann动力系统中著名的Moser变换的应用。它在$\mathbb{C}^{3n+1}$上给出了一个新的二次可积动力系统，我们称之为Neumann-Moser动力系统。我们给出了Moser变换逆变换的显式公式。因此，我们显式获得了一个可逆变换，将Neumann系统的Uhlenbeck-Devaney积分变换为我们系统的积分。本文的主要结果之一是Neumann-Moser系统的循环解。我们证明了我们系统的每个解都满足Mumford动力系统，反之亦然。Neumann-Moser系统的每个解都被证明满足稳态Korteweg-de Vries层级。作为推论，我们用超椭圆Kleinian函数构造了Neumann-Moser系统的显式解。

英文摘要：

At the focus of the paper are applications of the well-known Moser transformation of the C. Neumann dynamical system. It yields us a new quadratic integrable dynamical system on $\mathbb{C}^{3n+1}$, which we call the Neumann-Moser dynamical system. We present an explicit formula of the inverse of the Moser transformation. Consequently, we obtain explicitly an invertible transformation of the Uhlenbeck-Devaney integrals of the Neumann system into the integrals of our system. One of the main results of the paper is the recurrent solutions of the Neumann-Moser system. We show that every solution of our system solves the Mumford dynamical system, and vice versa. Every solution of the Neumann-Moser system is proven to solve the stationary Korteweg-de Vries hierarchy. As a corollary, we construct explicit solutions of the Neumann-Moser system in hyperelliptic Kleinian functions.

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