Application Of KAM Theory To Partial Differential Equations

In recent years,many new ideas and theories have been developed to study quasi-periodic solutions of PDEs(partial differential equations).For example,KAM theory is a very powerful tool developed in this process.The specific idea is to extend the classical KAM theory from finite-dimensional dynamic system to infinite-dimensional dynamic system.The first and second Melnikov non-resonance conditions are needed simultaneously in the Newton iteration process,and Birkhoff normal form is finally reduced near the invariant torus.By using Lyapunov Schmidt decomposition,CWB method only need the first Melnikov non-resonance condition during the Newton iteration process.However,the linear stability near the invariant KAM torus is not given,unlike the KAM method,which can prove the linear stability of the solution near the invariant torus.The study of PDEs in one-dimensional case and high-dimensional case is completely different.This is because the linear operator of the linear part of the equation has more complex eigenvalues and eigenfunctions in the high-dimensional case.So the first problem in this paper is to consider the existence of quasi-periodic solutions for the high-dimensionalquintic beam equation:(?)Many of the most concerned PDEs in physics are both Hamiltonian and reversible sys-tems.In fact,the partial differential equations arsing in physics not only have the derivatives in the nonlinearity,but also remove the Fourier multiplier or potential function from the lin-ear operator(that is,the case of complete resonance).In the first case,the regularity of the perturbation is weaker,which makes estimating the measure more difficult;in the sec-ond case,the simplified linear operator makes the complete resonance phenomenon more complex,which increase the difficulty of proving the small divisor condition,especially the second Melnikov non-resonance condition.So secondly we consider the existence of quasi-periodic solutions for the two-dimensional,reversible,completely resonant,and derivative nonlinear beam equations.(?)And thirdly we consider the existence of quasi-periodic solutions for two-dimensional,com-pletely resonant,and derivative Beam Equation which is a Hamiltonian system:(?)All above work is about PDEs with analytic pertubation.When the pertubation is no longer analytic,but Cm,i.e.,merely differentiable,the existent KAM theory could only deals with finite dimensional dynamic systems.For infinite dimensional dynamic systems there is no result via KAM method.Because in the analytic case,the chaotic region is exponential small,while in the finite differentiable case,the chaotic region is reducing with polynomial decay rate.Then the fourth question considered in this paper is studying the existence of time quasi-periodic breathers for the following simplified model of Newton's cradle lattice with Hertzian interaction:(?) is the Hertzian interaction potential.