Quasi-Periodic Solution of Nonlinear Beam Equation on $${\mathbb T}^d$$ T d with Forced Frequencies
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Quasi-Periodic Solution of Nonlinear Beam Equation on Td with Forced Frequencies Shidi Zhou1 Received: 24 June 2020 / Accepted: 9 September 2020 © Springer Nature Switzerland AG 2020
Abstract In this paper, we study the higher dimensional nonlinear beam equation under periodic boundary condition: ¯ u) = 0, u = u(t, x), t ∈ R, x ∈ Td , d ≥ 2 u tt + 2x u + Mξ u + f (ωt, ¯ u) is a real analytic function with where Mξ is a real Fourier multiplier, f = f (θ, respect to (θ¯ , u), and f (θ¯ , u) = O(|u|2 ). This equation can be viewed as an infinite dimensional nearly integrable Hamiltonian system. We establish an infinite dimensional KAM theorem, and apply it to this equation to prove that there exist a class of small amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori. Keywords KAM theory · Hamiltonian systems · Beam equation · Birkhoff normal form Mathematics Subject Classification Primary 37K55, 35B10
1 Introduction In this paper we consider the nonlinear beam equation in higher dimension: ¯ u) = 0, u = u(t, x), t ∈ R, x ∈ Td , d ≥ 2 (1.1) u tt + 2x u + Mξ u + f (ωt, ¯ u) is a real analytic function with where Mξ is a real Fourier multiplier, f = f (θ, ¯ u) = ¯ respect to (θ, u) in a neighborhood of the origin in Tm × R, and it satisfies f (θ,
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Shidi Zhou [email protected] School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai 519082, Guangdong, People’s Republic of China 0123456789().: V,-vol
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O(|u|2 ). ω¯ is a forced oscillation frequency. The parameter ξ ∈ O ⊆ Rb , where the parameter set O has positive Lebesgue measure. The forced frequency ω¯ ∈ Q ⊆ Rm , where the frequency set Q has positive Lebesgue measure. Here O and Q will both work in reaching the small divisor condition at each iterative step. Since the 1980s, the infinite dimensional KAM theory has attracted great interests. It has been proved to be a powerful tool in working with Hamiltonian PDEs. Since the remarkable pioneering work [8,22,23,34], many results about the existence of quasi-periodic solutions of one dimensional Hamiltonian PDEs have been got via KAM approach. For these works, see [7,12,16,20,21,23–28,35]. However, the higher dimensional Hamiltonian PDEs are quite different, because in this case, the linear operator of the equation has multiple eigenvalues. It is a big obstacle in working with the second Melnikov condition at each KAM iterative step. The first breakthrough came from Bourgain’s work [4] in 1998. In this remarkable paper, Bourgain proved the existence of a class of small-amplitude quasi-periodic solutions with two frequencies of a class of two-dimensional Schrödinger equation with a real Fourier multiplier. He used the method of multi-scale analysis to avoid the second Melnikov condition. Following the idea and method of [4], people have got a lot of results about higher dimensional equations, see [1–3,5,6,33]. Multi-scale analysis has been proved successful in working with higher dimensional Hamiltonian PDEs. However, it can’t describe the normal
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