KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation

被引:130
作者
Baldi, Pietro [1 ]
Berti, Massimiliano [1 ,2 ]
Montalto, Riccardo [2 ]
机构
[1] Univ Naples Federico II, Dipartimento Matemat & Applicaz R Caccioppoli, I-80126 Naples, Italy
[2] SISSA, I-34136 Trieste, Italy
基金
欧洲研究理事会;
关键词
PARTIAL-DIFFERENTIAL-EQUATIONS; PERIODIC-SOLUTIONS; WAVE-EQUATIONS; SINGULARITIES; THEOREM;
D O I
10.1007/s00208-013-1001-7
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
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.
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页码:471 / 536
页数:66
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