The Implementation Of The Third-order Pedigree And Its Constrained Flows Integrable Boussinesq System Involutive Solution

The evolution equations and the integrablity of the third order eigenvalue problem open problems. The key of realization is to find a real reasonable coordinate system. In this paper, we convert the complex third order eigenvalue problems into the real third order eigenvalue problems. Then, based on the Euler-Lagrange equation and Legendre transformation, a reasonable Jacobi-Ostrogredsky coordinate system have been found,then using nonlinear method,the Lax pairs of the real Bargrnann and Neumann system are nonlinearized, so as to be a new finite-dimensional integrable Hamilton system in the Liouville sense is generated. Moreover, the involutive representations of the solution for the evolution equations are obtained.