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:$$\begin{aligned} \begin{aligned} y_{tt}(t,x)=\mu y_{xxxx}+y_{xx}+\left( y^{3}+\varepsilon f(\omega t,x)\right) _{xx},\,\, x\in [0, \pi ],\,\, \mu >0, \end{aligned} \end{aligned}$$subject to the hinged boundary conditions$$\begin{aligned} \begin{aligned} y(t,0)=y(t,\pi )=y_{xx}(t,0)=y_{xx}(t,\pi )=0, \end{aligned} \end{aligned}$$where \(\omega =(1,\alpha )\) with \(\alpha \) 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 allow \(\alpha \) to be any irrational numbers, and thus the frequency \(\omega =(1,\alpha )\) 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.

中文翻译：

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Liouvillean频率拟周期强迫的不适定Boussinesq方程的响应解

在本文中，我们证明了拟周期强迫广义不适定Boussinesq方程的响应解（即与强迫频率相同的拟周期解）的存在：$$ \ begin {aligned} \ begin {aligned } y_ {tt}（t，x）= \ mu y_ {xxxx} + y_ {xx} + \ left（y ^ {3} + \ varepsilon f（\ omega t，x）\ right）_ {xx}， \，\，x \ in [0，\ pi]，\，\，\ mu> 0，\ end {aligned} \ end {aligned} $$受铰接边界条件$$ \ begin {aligned} \ begin {aligned} y（t，0）= y（t，\ pi）= y_ {xx}（t，0）= y_ {xx}（t，\ pi）= 0，\ end {aligned} \ end {aligned } $$其中\（\ omega =（1，\ alpha）\）与\（\ alpha \）是任何非理性数字。该证明基于改进的Kolmogorov-Arnold-Moser（KAM）迭代方案。我们将在KAM迭代的每个步骤中构造辛变换，以使这些变换的组成将原始系统还原为平衡为零的新系统。请注意，我们允许\（\ alpha \）为任何无理数，因此频率\（\ omega =（1，\ alpha）\）超出了丢丢丁或布儒诺频率，我们称其为“路易斯维南频率”。此外，所考虑的模型是不适定的，具有复杂的哈密顿结构。这使得出现在KAM迭代中的同构方程不同于经典的无穷维KAM理论中的方程。本文获得的结果加强了文献中现有的结果，在文献中该系统处于适当的位置或假定强迫频率为丢丢丁胺。