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- Title
Response Solution to Ill-Posed Boussinesq Equation with Quasi-Periodic Forcing of Liouvillean Frequency.
- Authors
Wang, Fenfen; Cheng, Hongyu; Si, Jianguo
- Abstract
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: y tt (t , x) = μ y xxxx + y xx + y 3 + ε f (ω t , x) xx , x ∈ [ 0 , π ] , μ > 0 , subject to the hinged boundary conditions y (t , 0) = y (t , π) = y xx (t , 0) = y xx (t , π) = 0 , where ω = (1 , α) with α 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 α to be any irrational numbers, and thus the frequency ω = (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.
- Subjects
BOUSSINESQ equations; IRRATIONAL numbers
- Publication
Journal of Nonlinear Science, 2020, Vol 30, Issue 2, p657
- ISSN
0938-8974
- Publication type
Article
- DOI
10.1007/s00332-019-09587-8