| Softcover ISBN: | 978-2-85629-892-3 |
| Product Code: | AST/403 |
| List Price: | $67.00 |
| AMS Member Price: | $53.60 |
| Softcover ISBN: | 978-2-85629-892-3 |
| Product Code: | AST/403 |
| List Price: | $67.00 |
| AMS Member Price: | $53.60 |
-
Book DetailsAstérisqueVolume: 403; 2018; 148 ppMSC: Primary 37; 35
The authors prove that small, semi-linear Hamiltonian perturbations of the defocusing nonlinear Schrödinger (dNLS) equation on the circle have an abundance of invariant tori of any size and (finite) dimension which support quasi-periodic solutions. When compared with previous results, the novelty consists in considering perturbations which do not satisfy any symmetry condition (they may depend on \(x\) in an arbitrary way) and need not be analytic. The main difficulty is posed by pairs of almost resonant dNLS frequencies.
The proof is based on the integrability of the dNLS equation, in particular, the fact that the nonlinear part of the Birkhoff coordinates is one smoothing.
The authors implement a Newton-Nash-Moser iteration scheme to construct the invariant tori. The key point is the reduction of linearized operators, coming up in the iteration scheme, to \(2 \times 2\) block diagonal ones with constant coefficients together with sharp asymptotic estimates of their eigenvalues.
A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.
ReadershipGraduate students and research mathematicians.
-
Additional Material
-
RequestsReview Copy – for publishers of book reviewsAccessibility – to request an alternate format of an AMS title
- Book Details
- Additional Material
- Requests
The authors prove that small, semi-linear Hamiltonian perturbations of the defocusing nonlinear Schrödinger (dNLS) equation on the circle have an abundance of invariant tori of any size and (finite) dimension which support quasi-periodic solutions. When compared with previous results, the novelty consists in considering perturbations which do not satisfy any symmetry condition (they may depend on \(x\) in an arbitrary way) and need not be analytic. The main difficulty is posed by pairs of almost resonant dNLS frequencies.
The proof is based on the integrability of the dNLS equation, in particular, the fact that the nonlinear part of the Birkhoff coordinates is one smoothing.
The authors implement a Newton-Nash-Moser iteration scheme to construct the invariant tori. The key point is the reduction of linearized operators, coming up in the iteration scheme, to \(2 \times 2\) block diagonal ones with constant coefficients together with sharp asymptotic estimates of their eigenvalues.
A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.
Graduate students and research mathematicians.
