KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation
KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation
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DOI：
10.1007/s00208-013-1001-7
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发表时间：
2014-01
影响因子：
1.4
通讯作者：
P. Baldi;M. Berti;Riccardo Montalto
中科院分区：
文献类型：
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作者：
P. Baldi;M. Berti;Riccardo Montalto
我们证明了线性艾里方程的准线性和完全非线性受迫扰动的小振幅准周期解的存在性。对于哈密顿或可逆非线性，我们还证明了它们的线性稳定性。关键分析涉及线性化算子在近似解处的可约性，它提供了其特征值的急剧渐近展开。对于准线性扰动，这不能通过 KAM 迭代直接获得。因此，我们首先执行正则化过程，将线性化运算符与具有常数系数和有界余数的运算符共轭。这些变换是通过环面和伪微分算子的微分同胚引起的变量变化来获得的。此时，我们实施纳什-莫泽迭代（具有二阶梅尔尼科夫非共振条件），完成对常数系数的简化。
We prove the existence of small amplitude quasi-periodic solutions forquasi-linearandfully nonlinearforced perturbations of the linear Airy equation. For Hamiltonian or reversible nonlinearities we also prove their linear stability. The key analysis concerns the reducibility of the linearized operator at an approximate solution, which provides a sharp asymptotic expansion of its eigenvalues. For quasi-linear perturbations this cannot be directly obtained by a KAM iteration. Hence we first perform a regularization procedure, which conjugates the linearized operator to an operator with constant coefficients plus a bounded remainder. These transformations are obtained by changes of variables induced by diffeomorphisms of the torus and pseudo-differential operators. At this point we implement a Nash–Moser iteration (with second order Melnikov non-resonance conditions) which completes the reduction to constant coefficients.