De Toda à KdV

摘要：

我们考虑了周期Toda格点的大粒子数极限，其初始数据族接近于平衡态。我们表明，相应的雅可比矩阵的频谱的两个边缘的每一个是一个错误，由两个希尔运营商的频谱，与这个家庭。然后，我们表明，雅可比矩阵的谱几乎保持不变时，矩阵沿两个限制KdV方程的演变。最后，我们证明了Toda的行动，当适当的重整化，收敛到KdV的。

英文摘要：

We consider the large number of particles limit of a periodic Toda lattice for a family of initial data close to the equilibrium state. We show that each of the two edges of the spectra of the corresponding Jacobi matrices is up to an error, determined by the spectra of two Hill operators, associated to this family. We then show that the spectra of the Jacobi matrices remain almost constant when the matrices evolve along the two limiting KdV equations. Finally we prove that the Toda actions, when appropriately renormalized, converge to the ones of KdV .

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