Open AccessAction-angle variables for the Lie-Poisson Hamiltonian systems associated with the three-wave resonant interaction system
Author(s) -
Xue Geng,
L. Guan,
Dianlou Du
Publication year - 2022
Publication title -
aims mathematics
Language(s) - English
Resource type - Journals
ISSN - 2473-6988
DOI - 10.3934/math.2022557
Subject(s) - integrable system , eigenfunction , poisson distribution , hamiltonian (control theory) , hamiltonian system , mathematical physics , mathematics , separation of variables , mathematical analysis , lie algebra , pure mathematics , physics , quantum mechanics , eigenvalues and eigenvectors , partial differential equation , mathematical optimization , statistics
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.