The Twice Integrable Rosochatius Deformations Of The Restricted Soliton Flows

Since most of Hamiltonian systems are not integrable, it is a challenging work toconstruct new integrable Hamiltonian systems. It is a natural idea to generate new in-tegrable systems by perturbation. Recently, an approach called integrable Rosochatiusdeformation has been proposed and applied to quite a lot of restricted soliton flows. Inparticular, the Rosochatius deformations of the geodesic flow equation on the ellipsoidand the restricted mKdV flow have been constructed. In this thesis, we further developthe approach to constructing twice Rosochatius deformations of restricted soliton flows.The thesis consists of three chapters.First of all, we briefly summarize the history of Rosochatius deformations of finitedimensional integrable Hamiltonian systems. Then in the second chapter, the twiceRosochatius deformation of the geodesic flow equation on the ellipsoid is constructed.A new integrable Hamiltonian system defined on a new surface is obtained. It is shownthat the obtaining system enjoys a Lax representation and the Lax matrix satisfies anew r-matrix relation. A set of its conversed integrals of motion are presented. Thethird chapter is devote to the twice Rosochatius deformation of the restricted mKdVflow. A new free integrable Hamiltonian system is generated. Its Lax representationand r-matrix relation, as well as the first integrals are derived.