Response solutions to ill-posed Boussinesq equation with quasi-periodic forcing of Liouvillean frequency
Response solutions to ill-posed Boussinesq equation with quasi-periodic forcing of Liouvillean frequency
复制标题
具有刘维尔频率准周期强迫的不适定 Boussinesq 方程的响应解
DOI：
10.1007/s00332-019-09587-8
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发表时间：
2019
影响因子：
3
通讯作者：
Jianguo Si
中科院分区：
文献类型：
--
作者：
Fenfen Wang;Hongyu Cheng;Jianguo Si
本文证明了拟周期强迫广义不适定Boussinesq方程：$$\Begin{Align}\Begin{Align}y_{tt}(t，x)=\muy_{xxxx}+y_{xx}+\Left(y^{3}+\varepsilon f(\omega t，x)\right)_{xx}，\，\，x\in[0，\pi]，\，\，\mU>；0，\结束{对齐}\结束{对齐}$$ytt(t，x)=μyxxxx+yxx+y3+εf(ωt，x)xx，x∈[0，π]，μ>；0，满足铰链边界条件$$\Begin{Aligned}y(t，0)=y(t，\pi)=y_{xx}(t，0)=y_{xx}(t，\pi)=0，\end{Align}\end{Align}$$y(t，0)=y(t，π)=yxx(t，0)=yxx(t，π)=0，其中ω=(1，α)，α为任意无理数。证明基于修正的Kolmogorov-Arnold-Moser(KAM)迭代格式。在KAM迭代的每一步，我们将构造一个辛变换，使得这些变换的组合将原始系统归结为一个以零为平衡的新系统。注意，我们允许α是任何无理数，因此频率ω=(1，α)超过丢番图或布鲁朱诺频率，我们称之为刘维利频率。此外，所考虑的模型是不适定的，并且具有复杂的哈密顿结构。这使得在KAM迭代中出现的同调方程不同于经典无限维KAM理论中的同调方程。本文的结果加强了文献中系统适定性或强迫频率为丢番图的已有结果。
In this paper, we prove the existence of response solution (i.e., quasi-periodic solution with the same frequency as the forcing) for the quasi-periodically forced generalized ill-posed Boussinesq equation: $$\begin{aligned} \begin{aligned} y_{tt}(t,x)=\mu y_{xxxx}+y_{xx}+\left( y^{3}+\varepsilon f(\omega t,x)\right) _{xx},\,\, x\in [0, \pi ],\,\, \mu >0, \end{aligned} \end{aligned}$$ytt(t,x)=μyxxxx+yxx+y3+εf(ωt,x)xx,x∈[0,π],μ>0,subject to the hinged boundary conditions $$\begin{aligned} \begin{aligned} y(t,0)=y(t,\pi )=y_{xx}(t,0)=y_{xx}(t,\pi )=0, \end{aligned} \end{aligned}$$y(t,0)=y(t,π)=yxx(t,0)=yxx(t,π)=0,whereω=(1,α) withα being any irrational numbers. The proof is based on a modified Kolmogorov–Arnold–Moser (KAM) iterative scheme. We will, at every step of KAM iteration, construct a symplectic transformation in a such way that the composition of these transformations reduce the original system to a new system which possesses zero as equilibrium. Note that we allowα to be any irrational numbers, and thus the frequencyω=(1,α) is beyond Diophantine or Brjuno frequency, which we call as Liouvillean frequency. Moreover, the model under consideration is ill-posed and has complicated Hamiltonian structure. This makes homological equations appearing in KAM iteration are different from the ones in the classical infinite-dimensional KAM theory. The result obtained in this paper strengthens the existing results in the literature where the system is well-posed or the forcing frequency is assumed to be Diophantine.