A finite-dimensional integrable system associated with the three-wave interaction equations

Under a constraint between the potentials and the eigenfunctions, the 3 × 3 AKNS matrix spectral problem and its adjoint spectral problem associated with the three-wave interaction equations are nonlinearized so as to be a new finite-dimensional Hamiltonian system. A general scheme for generating involutive systems of conserved integrals and their two new generators are proposed, by which the finite-dimensional Hamiltonian system is further proved to be completely integrable in the Liouville sense. Moreover, the involutive solutions of the three-wave interaction equations are given.

This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Yongtang Wu, Xianguo Geng; A finite-dimensional integrable system associated with the three-wave interaction equations. J. Math. Phys. 1 July 1999; 40 (7): 3409–3430. https://doi.org/10.1063/1.532896 and may be found at https://pubs.aip.org/aip/jmp/article/40/7/3409/231271/A-finite-dimensional-integrable-system-associated.