KAM Tori for a Two Dimensional Beam Equation with a Quintic Nonlinear Term and Quasi-periodic Forcing
Tóm tắt
This work studies a two-dimensional beam equation with a quintic nonlinear term and quasi-periodic forcing
$$\begin{aligned} u_{tt}+\Delta ^2 u+ \varepsilon \phi (t)h(u)=0,\quad x\in {\mathbb {T}}^2,\quad t\in {\mathbb {R}} \end{aligned}$$
with periodic boundary conditions, where
$$\varepsilon $$
is a small positive parameter;
$$\phi (t)$$
is a real analytic quasi-periodic function in t with frequency vector
$$\eta =(\eta _1 ,\eta _2 \ldots ,\eta _{n^*})\subset [\varrho , 2\varrho ]^{n^*}$$
for a given positive integer
$$n^*$$
and some constant
$$\varrho >0$$
; and h is a real analytic function of the form
$$\begin{aligned} h(u)=c_1u+c_5u^5+\sum _{{\hat{i}}\ge 6}c_{{\hat{i}}}u^{{\hat{i}}},\quad c_1,c_5\ne 0. \end{aligned}$$
Firstly, the linear part of Hamiltonian system corresponding to the equation is transformed 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 the appropriate tangential sites. Finally, it is obtained that a Whitney smooth family of small-amplitude quasi-periodic solutions for the equation by developing an abstract KAM (Kolmogorov–Arnold–Moser) theorem for infinite dimensional Hamiltonian systems.
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