KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation
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- 作者：Pietro Baldi (1)
Massimiliano Berti (1) (2)
Riccardo Montalto (2) - 关键词：37K55 ; 35Q53
- 刊名：Mathematische Annalen
- 出版年：2014
- 出版时间：June 2014
- 年：2014
- 卷：359
- 期：1-2
- 页码：471-536
- 全文大小：
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Massimiliano Berti (1) (2)
Riccardo Montalto (2)
1. Dipartimento di Matematica e Applicazioni “R. Caccioppoli- Università degli Studi Napoli Federico II, Via Cintia, Monte S. Angelo, 80126?, Naples, Italy
2. SISSA, Via Bonomea 265, 34136?, Trieste, Italy - ISSN：1432-1807
文摘
We prove the existence of small amplitude quasi-periodic solutions for quasi-linear and fully nonlinear forced 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.