Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation
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.
scientific article; zbMATH DE number 854852