We present recent results concerning quasi-periodic solutions for quasilinear 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.

A note on KAM theory for quasi-linear and fully nonlinear forced KdV / Baldi, P.; Berti, Massimiliano; Montalto, Riccardo. - In: ATTI DELLA ACCADEMIA NAZIONALE DEI LINCEI. RENDICONTI LINCEI. MATEMATICA E APPLICAZIONI. - ISSN 1720-0768. - 24:3(2013), pp. 437-450. [10.4171/RLM/660]

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

Berti, Massimiliano;Montalto, Riccardo
2013-01-01

Abstract

We present recent results concerning quasi-periodic solutions for quasilinear 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.
2013
24
3
437
450
http://www.ems-ph.org/journals/show_abstract.php?issn=1120-6330&vol=24&iss=3&rank=5
http://preprints.sissa.it/xmlui/handle/1963/7232
Baldi, P.; Berti, Massimiliano; Montalto, Riccardo
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/17449
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