Sobolev stability of plane wave solutions to the cubic nonlinear Schrödinger equation on a torus
Please always quote using this URN:urn:nbn:de:0296-matheon-11881
- It is shown that plane wave solutions to the cubic nonlinear Schrödinger equation on a torus behave orbitally stable under generic perturbations of the initial data that are small in a high-order Sobolev norm, over long times that extend to arbitrary negative powers of the smallness parameter. The perturbation stays small in the same Sobolev norm over such long times. The proof uses a Hamiltonian reduction and transformation and, alternatively, Birkhoff normal forms or modulated Fourier expansions in time.
Author: | Erwan Faou, Ludwig Gauckler, Christian Lubich |
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URN: | urn:nbn:de:0296-matheon-11881 |
Referee: | Reinhold Schneider |
Document Type: | Preprint, Research Center Matheon |
Language: | English |
Date of first Publication: | 2013/11/06 |
Release Date: | 2013/11/06 |
Tag: | nonlinear Schrödinger equation; plane wave; stability |
Institute: | Research Center Matheon |
Technische Universität Berlin | |
MSC-Classification: | 35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Bxx Qualitative properties of solutions / 35B35 Stability |
35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Qxx Equations of mathematical physics and other areas of application [See also 35J05, 35J10, 35K05, 35L05] / 35Q55 NLS-like equations (nonlinear Schrödinger) [See also 37K10] | |
37-XX DYNAMICAL SYSTEMS AND ERGODIC THEORY [See also 26A18, 28Dxx, 34Cxx, 34Dxx, 35Bxx, 46Lxx, 58Jxx, 70-XX] / 37Kxx Infinite-dimensional Hamiltonian systems [See also 35Axx, 35Qxx] / 37K55 Perturbations, KAM for infinite-dimensional systems | |
Preprint Number: | 1042 |