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

Erwan Faou (1, 2) , Ludwig Gauckler (3) , Christian Lubich (4)

1 IPSO - Invariant Preserving SOlvers
2 IRMAR - Institut de Recherche Mathématique de Rennes
3 Institut für Mathematik [Berlin]
4 Mathematisches Institut [Tübingen]

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.

Mots clés: Nonlinear Schrödinger equation; Normal form; Modulated Fourier expansion

Domaines: Equations aux dérivées partielles [math.AP]

Erwan Faou, Ludwig Gauckler, Christian Lubich. Sobolev stability of plane wave solutions to the cubic nonlinear Schrödinger equation on a torus. Communications in Partial Differential Equations, 2013, 38 (7), pp.1123-1140. ⟨10.1080/03605302.2013.785562⟩. ⟨hal-00622240v2⟩