KAM Theory And Higher Dimensional Hamiltonian PDE

In this thesis,we mainly study infinite dimensional KAM theory with applications to some classes of higher dimensional Hamiltonian PDEs.We concentrate on two important classes of Hamiltonian PDEs:two-dimensional completely resonant beam equation,and higher dimensional Schrodinger system.By dealing with the Birkhoff normal form of the equation,and establishing an abstract infinite dimensional KAM theorem,we prove that,in the sense of measure,there exists abundant small-amplitude quasi-periodic solutions of the corresponding equations.In the first chapter,on one hand,we state the classical KAM theory,and on the other hand,we talk about the main interests in studying Hamiltonian PDEs.In addition,we review the main results in recent years.In the second chapter,we study the existence of quasi-periodic solution of the two-dimensional completely resonant beam equation under periodic boundary condition.At first,we deal with the Birkhoff normal form of the equation and we could eliminate the nonresonant terms in the normal form by a canonical symplectic transformation.Then by making use of the zero-momentum condition and the special structure of the admissible set,we could prove that each normal variable corresponding to each single integer point could only appear in at most one non-integrable term in the normal form after the transformation.So we could get a normal form with the form of block-diagonal,with each block of degree at most 2×2.In addition,this normal form is dependent on the angle variable.Then we could apply the abstract KAM theorem to conduct infinite many steps of iteration,and we could get the quasi-periodic solution by proving that the iterative sequences converge.During the KAM iteration,the parameters are provided by the amplitude of the solution.We need to say that because our normal form is dependent on the angle variable,we couldn't get the linear stability at last.In the third charpter,we study the existence of quasi-periodic solutions of non-linear higher dimensional Schrodinger system.The parameters are provided by the artificial Fourier Multiplier.At first,we deal with the normal form of the equation.Here because the two equations are coupled,some coupled non-integrable terms also appear in the normal form.The normal form is still of the form of block-diagonal with each block of degree at most 2×2.Then we could apply the abstract KAM theorem to conduct infinite many steps of iteration,and we could get the quasi-periodic solution by proving that the iterative sequences converge.Here due to the absence of regularity of the nonlinearity,we need to apply the Toplitz-Lipschitz property to conduct the measure estimate.