Poisson commuting energies for a system of infinitely many bosons
Poisson commuting energies for a system of infinitely many bosons
复制标题
DOI：
10.1016/j.aim.2022.108525
复制
发表时间：
2019-10
影响因子：
1.7
通讯作者：
Dana Mendelson;A. Nahmod;Natavsa Pavlovi'c;M. Rosenzweig;G. Staffilani
中科院分区：
文献类型：
--
作者：
Dana Mendelson;A. Nahmod;Natavsa Pavlovi'c;M. Rosenzweig;G. Staffilani
我们认为立方Gross-Pitaevskii（GP）在一个空间维度的层次。我们建立了一个无穷序列的可观量，使相应的迹泛函，我们称之为“能量”，交换关于弱李泊松结构的作者在[57]中定义。与第三能量泛函相关联的哈密顿方程正是GP族。对应于剩余能量的运动方程推广了著名的非线性薛定谔方程组，其中第三个元素是一维立方非线性薛定谔方程。这项工作提供了大量的证据，GP的层次作为一个新的可积系统，是一个步骤，了解的起源的可积性的NLS在一个量子可积系统的标度极限。
We consider the cubic Gross-Pitaevskii (GP) hierarchy in one spatial dimension. We establish the existence of an infinite sequence of observables such that the corresponding trace functionals, which we call “energies,” commute with respect to the weak Lie-Poisson structure defined by the authors in [57]. The Hamiltonian equation associated to the third energy functional is precisely the GP hierarchy. The equations of motion corresponding to the remaining energies generalize the well-known nonlinear Schrödinger hierarchy, the third element of which is the one-dimensional cubic nonlinear Schrödinger equation. This work provides substantial evidence for the GP hierarchy as a new integrable system and is a step towards understanding the origins of the integrability of the NLS in terms of a scaling limit of a quantum integrable system.