Resumen de KAM Tori for a Two Dimensional Beam Equation with aQuintic Nonlinear Term and Quasi-periodic Forcing
Min Zhang, Jie Rui, Yan Li, Jian Zhang
This work studies a two-dimensional beam equation with a quintic nonlinear term andquasi-periodic forcingutt+2u+εφ(t)h(u)=0,x∈T2,t∈Rwith periodic boundary conditions, whereεis a small positive parameter;φ(t)is a realanalytic quasi-periodic function intwith frequency vectorη=(η1,η2...,ηn∗)⊂[,2]n∗for a given positive integern∗and some constant>0; andhis a realanalytic function of the formh(u)=c1u+c5u5+∑ˆi≥6cˆiuˆi,c1,c5=0.Firstly, the linear part of Hamiltonian system corresponding to the equation is trans-formed to constant coefficients by a linear quasi-periodic change of variables. Then,a symplectic transformation is used to convert the Hamiltonian system into an angle-dependent block-diagonal normal form, which can be achieved by selecting theappropriate tangential sites. Finally, it is obtained that a Whitney smooth family ofsmall-amplitude quasi-periodic solutions for the equation by developing an abstractKAM (Kolmogorov–Arnold–Moser) theorem for infinite dimensional Hamiltonian systems.
