Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation. (English) Zbl 0847.35130

Kuksin, Sergej; Pöschel, Jürgen

Ann. Math. (2) 143, No. 1, 149-179 (1996).

This paper deals with the nonlinear Schrödinger equation \[ iu_t= u_{xx}- mu- f(|u|^2) u \] on \([0, T]\) with Dirichlet boundary conditions. After the introduction which contains the main results, the paper continues as follows. The Hamiltonian of the nonlinear equation is written in infinitely many coordinates, and its regularity is established. And then it is transformed into its Birkhoff normal form of order four. In the next section, another theorem is stated about the existence of invariant Cantor manifolds for Hamiltonians. This result is used to prove the basic theorem of the paper. The next result is reduced to a KAM-theorem regarding perturbations of families of linear Hamiltonians. Shortly, to summarize, the paper provides a number of small amplitude solutions which are quasi-periodic in time.

Reviewer: G.Jumarie (Montreal)

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
37J35	Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10	Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35B05	Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs

Keywords:

nonlinear Schrödinger equation; invariant Cantor manifolds; Hamiltonian; KAM-theorem; small amplitude solutions