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On the Integrability of the Benjamin-Ono Equation on the Torus
Communications on Pure and Applied Mathematics ( IF 3 ) Pub Date : 2020-04-22 , DOI: 10.1002/cpa.21896
Patrick Gérard 1 , Thomas Kappeler 2
Affiliation  

In this paper we prove that the Benjamin-Ono equation, when considered on the torus, is an integrable (pseudo)differential equation in the strongest possible sense: it admits global Birkhoff coordinates on the space $L^2(\T)$. These are coordinates which allow to integrate it by quadrature and hence are also referred to as nonlinear Fourier coefficients. As a consequence, all the $L^2(\T)$ solutions of the Benjamin--Ono equation are almost periodic functions of the time variable. The construction of such coordinates relies on the spectral study of the Lax operator in the Lax pair formulation of the Benjamin--Ono equation and on the use of a generating functional, which encodes the entire Benjamin--Ono hierarchy.

中文翻译:

关于圆环上 Benjamin-Ono 方程的可积性

在本文中,我们证明了 Benjamin-Ono 方程,当考虑到环面时,是一个最强意义上的可积(伪)微分方程:它承认空间 $L^2(\T)$ 上的全局 Birkhoff 坐标。这些是允许通过正交对其进行积分的坐标,因此也称为非线性傅立叶系数。因此,本杰明-小野方程的所有 $L^2(\T)$ 解几乎都是时间变量的周期函数。此类坐标的构建依赖于对 Benjamin-Ono 方程的 Lax 对公式中的 Lax 算子的谱研究,以及对整个 Benjamin-Ono 层次结构进行编码的生成泛函的使用。
更新日期:2020-04-22
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