Existence Of Quasi-periodic Solutions For The First-Order Approximation Boussinesq Equation

In this paper,a one-dimensional the first-order approximation Boussinesq equation with periodic boundary conditions is considered.It is proved that the above equation admits small-amplitude quasi-periodic solutions corresponding to linearly stable n dimensional elliptic diophantine invariant tori of an associated infinite dimensional Hamiltonian system.The proof is based on the Cantor manifold theorem and the Birkhoff normal form.The thesis is mainly consisted of four chapters.Some relevant theories of Hamiltonian system are recalled briefly in the first part,formulating an infinite dimensional KAM theorem and the Cantor manifold theorem.Later,it is nec-essary to describe the Hamiltonian formalism of the first-order approximation Boussinesq equation,the main theorem would be stated.In the third section,the fourth-order Birkhoff normal form can be obtained from Hamiltonian.The main result is proven by Cantor manifold theorem in chapter Ⅳ.By the way,the notorious and unpleasant calculations are referred to appendix.