A note on KAM theory for quasi-linear and fully nonlinear forced KdV

作者：Baldi Pietro^*; Berti Massimiliano; Montalto Riccardo

来源：Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica e Applicazioni, 2013, 24(3): 437-450.

DOI：10.4171/RLM/660

• KdV
• quasi-linear and fully nonlinear PDEs
• quasi-periodic solutions
• KAM for PDEs
• Nash-Moser theory

We present the recent results in [3] concerning quasi-periodic solutions for quasi-linear and fully nonlinear forced perturbations of KdV equations. For Hamiltonian or reversible nonlinearities the solutions are linearly stable. The proofs are based on a combination of different ideas and techniques: (i) a Nash-Moser iterative scheme in Sobolev scales. (ii) A regularization procedure, which conjugates the linearized operator to a differential 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. (iii) A reducibility KAM scheme, which completes the reduction to constant coefficients of the linearized operator, providing a sharp asymptotic expansion of the perturbed eigenvalues.

• 出版日期2013