Sobolev stability of plane wave solutions to the cubic nonlinear Schrödinger equation on a torus

Details Sobolev stability of plane wave solutions to the cubic nonlinear Schrödinger equation on a torus Sobolev stability of plane wave solutions to the cubic nonlinear Schrödinger equation on a torus

2011-09-12

arxiv.org/abs/1109.2407v1

Mathematics

Analysis of PDEs

Scientific paper

It is shown that plane wave solutions to the cubic nonlinear Schr\"odinger 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.

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