论文标题
Benjamin的完整整合性 - 多索氏歧管上的方程式
Complete integrability of the Benjamin--Ono equation on the multi-soliton manifolds
论文作者
论文摘要
本文致力于证明本杰明（Benjamin-ono（bo）方程式）在线上的完全集成性，仅限于每一个$ n $ soliton歧管，并由$ \ mathcal {u} _n $表示。我们构建广义动作 - 角度坐标，该坐标从$ \ Mathcal {u} _n $中建立真实的分析符号呈现，以在$ \ mathbb {r}^{2n} $的某些开放式凸子集上，并允许任何此类初始数据来求解方程式。结果，$ \ MATHCAL {U} _n $是Gérard-kappeler $ [19] $的bo equation $ n $ gap电位的普遍覆盖。 $ \ Mathcal {u} _n $中BO方程的全局良好性是通过多项式表征和歧管$ \ Mathcal {U} _n $的光谱表征给出的。除了对BO方程的LAX运算符和作用于某些耐寒空间的Shift Semigroup的光谱分析外，此类坐标的构建还依赖于使用生成功能的使用，该功能编码整个BO层次结构。
This paper is dedicated to proving the complete integrability of the Benjamin--Ono (BO) equation on the line when restricted to every $N$-soliton manifold, denoted by $\mathcal{U}_N$. We construct generalized action--angle coordinates which establish a real analytic symplectomorphism from $\mathcal{U}_N$ onto some open convex subset of $\mathbb{R}^{2N}$ and allow to solve the equation by quadrature for any such initial datum. As a consequence, $\mathcal{U}_N$ is the universal covering of the manifold of $N$-gap potentials for the BO equation on the torus as described by Gérard--Kappeler $[19]$. The global well-posedness of the BO equation in $\mathcal{U}_N$ is given by a polynomial characterization and a spectral characterization of the manifold $\mathcal{U}_N$. Besides the spectral analysis of the Lax operator of the BO equation and the shift semigroup acting on some Hardy spaces, the construction of such coordinates also relies on the use of a generating functional, which encodes the entire BO hierarchy.
