三次非线性薛定谔方程的Atiyah-Hitchin括号二.周期势
The Atiyah-Hitchin bracket for the cubic nonlinear Schrodinger equation. ii. Periodic potentials
摘要：
这是第二次在一系列文件的泊松形式主义的立方非线性薛定谔方程排斥非线性。在本文中，我们考虑周期势。周期辅助Dirac算子的逆谱问题导致一个超椭圆Riemann曲面$\G$。利用谱问题，我们在这个黎曼曲面上引入了一个亚纯函数$¶$。我们称之为外尔函数，因为它与第一篇论文中讨论的经典外尔函数密切相关。我们表明，对$(\G,¶)$进行自然的泊松结构。我们称之为畸形阿蒂亚希钦括号相空间上的Poisson括号是变形的Atiyah-Hitchin括号在逆谱变换下的图像。
英文摘要：
This is the second in a series of papers on Poisson formalism for the cubic nonlinear Schrödinger equation with repulsive nonlinearity. In this paper we consider periodic potentials. The inverse spectral problem for the periodic auxiliary Dirac operator leads to a hyperelliptic Riemann surface $\G$. Using the spectral problem we introduce on this Riemann surface a meromorphic function $¶$. We call it the Weyl function, since it is closely related to the classical Weyl function discussed in the first paper. We show that the pair $(\G,¶)$ carries a natural Poisson structure. We call it the deformed Atiyah--Hitchin bracket. The Poisson bracket on the phase space is the image of the deformed Atiyah--Hitchin bracket under the inverse spectral transform.
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