Quasi-periodic solutions for a class of beam equation system

Abstract

In this paper, we establish an abstract infinite dimensional KAM theorem. As an application, we use the theorem to study the higher dimensional beam equation system    u 1tt + ∆ 2 u 1 + σu 1 + u 1 u 2 2 = 0 u 2tt + ∆ 2 u 2 + µu 2 + u 2 1 u 2 = 0 under periodic boundary conditions, where 0 < σ ∈ [σ 1 , σ 2 ], 0 < µ ∈ [µ 1 , µ 2 ] are real parameters. By establishing a block-diagonal normal form, we obtain the existence of a Whitney smooth family of small amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional dynamic system.

Analogues of Kahan’s Method for Higher Order Equations of Higher Degree

“…In the linear case, the model was considered recently from the viewpoint of a Hamilton-Jacobi approach to higher order implicit systems [8], while a coupled PDE system of beam equations with cubic nonlinearity was analysed in [27]. From the second order Lagrangian (21), we can introduce the Ostrogradsky variables (see [1], for instance), given by…”

Analogues of Kahan's method for higher order equations of higher degree

“…In the linear case, the model was considered recently from the viewpoint of a Hamilton-Jacobi approach to higher order implicit systems [8], while a coupled PDE system of beam equations with cubic nonlinearity was analysed in [27]. From the second order Lagrangian (4.1), we can introduce the Ostrogradsky variables (see [1], for instance), given by…”