On Commuting Hamiltonians And Arnold Diffusion In A Lattice System

Recently, many mathematicians begin to study the diffusion in Hamilto-nian PDEs, for example, Schrodinger equation, wave equation, Kdv equation, these all can be treated as infinite degrees of freedom Hamiltonian systems. A recent progress for the cubic deforcusing nonlinear Schrodinger equation has been made in [], where,"diffusion" is exhibited by smooth solutions for which the energy moves from lower order Fourier modes to higher order Fourier modes. However, one does not know whether such transition of energy persists as time goes to infinity. For other PDEs, we still don’t know whether there exist similar diffusion phenomena. An effective way to study the diffusion in Hamiltonian PDEs is to study the lattice system first. In [], he proved in a discrete Schrodinger equation, there exists Arnold diffusion, but the pertur-bation is so special that all the periodic orbits are preserved, so the transition chain is preserved, essentially, it is of Arnold’s type. In [], they proved that for a lattice of perturbation of couple pendulums, there exists diffusion and the speed is estimated. In this model, the lattice system can be considered as a discretness of the sine-Gorden equation, the pendulums are freely connected in the unperturbed system.In the third section of this thesis, we studied the dynamics of two Hamil-tonians which are commutative from the variational point of view. We proved that the corresponding Aubry set, Mane set are the same separately, the corre-sponding Lax-Oleinik semigroup are commutative, they have a common viscosity subsolution of C1,1, and the a function is quasi-linear. In the fourth section of this thesis, we focus on studying a periodic lattice system which can be thought as N particals with springs connecting from the first one to the last one, and the last one is also connected with the first one. In the unperturbed system, these springs are not free. Through the concrete study of the Mather set, Aubry set, Mane set of this model, and the location of the invariant cylinder, we find a reasonable path which the diffusion can occur, using the variational method, we proved the existence of the diffusion orbit. By the regularity of the barrier functions in [], we needn’t use the c-equivalence to construct the local(global) connecting orbits, the diffusion orbits we construct obey the Arnold mechanism, this is different from the one constructed in [], where part of each diffusion orbit walks along the invariant cylinder. In fact, the mechanism of Arnold diffusion can be Arnold mechanism, or non-Arnold mechanism, from this we can see that the dynamics of multi-degrees of freedom of Hamiltonian system are very complicated.