Action-angle variables for the Lie-Poisson Hamiltonian systems associated with the three-wave resonant interaction system
Action-angle variables for the Lie-Poisson Hamiltonian systems associated with the three-wave resonant interaction system
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DOI：
10.3934/math.2022557
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发表时间：
2022
期刊：
影响因子：
2.2
通讯作者：
Xue Geng;L. Guan;Dianlou Du
中科院分区：
文献类型：
--
作者：
Xue Geng;L. Guan;Dianlou Du
应用由 $ 3\times 3 $ Lax 矩阵控制的 $ \mathfrak{gl}_3(\mathbb{C}) $ 有理高丁模型来研究势能和本征函数之间约束下的三波共振相互作用系统（TWRI）。并将TWRI系统分解为两个有限维李-泊松哈密顿系统。基于守恒积分的生成函数，证明了两个有限维李-泊松哈密顿系统在刘维尔意义上完全可积。通过 Sklyanin 变量分离方法计算与非超椭圆谱曲线相关的作用角变量，并分析与所得有限维可积李-泊松哈密顿系统和三波共振相互作用系统相关的雅可比反演问题。
The $ \mathfrak{gl}_3(\mathbb{C}) $ rational Gaudin model governed by $ 3\times 3 $ Lax matrix is applied to study the three-wave resonant interaction system (TWRI) under a constraint between the potentials and the eigenfunctions. And the TWRI system is decomposed so as to be two finite-dimensional Lie-Poisson Hamiltonian systems. Based on the generating functions of conserved integrals, it is shown that the two finite-dimensional Lie-Poisson Hamiltonian systems are completely integrable in the Liouville sense. The action-angle variables associated with non-hyperelliptic spectral curves are computed by Sklyanin's method of separation of variables, and the Jacobi inversion problems related to the resulting finite-dimensional integrable Lie-Poisson Hamiltonian systems and three-wave resonant interaction system are analyzed.