Perturbations of the defocusing nonlinear Schrödinger equation. (English) Zbl 1048.37067

The authors study the KAM theory for the defocusing nonlinear Schrödinger (NLS) equation \[ i\partial_t\varphi=-\partial^2_x\varphi+2| \varphi| ^2\varphi, \quad \varphi(x+1,t)=\varphi(x,t), \quad x,t\in {\mathbb{R}}. \] The basic theorem says that many of the NLS-invariant tori (not necessarily close to the zero solution) persist and remain linearly stable under small perturbations of the Hamiltonian satisfying three natural conditions. Main technical tool is a detailed treatment of the frequencies \(\omega_k\). In particular, new closed formulas for \(\omega_k\) are provided which are of independent interest.
The purpose of the present paper is to document that some results and methods from T. Kappeler and J. Pöschel [KdV & KAM, Berlin: Springer (2003; Zbl 1032.37001)] can applied to the NLS equation, too.
The main results of the paper were previously announced by the authors in [J. Nonlinear Math. Phys. 8, Suppl., 133–138 (2001; Zbl 0977.35133].


37K55 Perturbations, KAM theory for infinite-dimensional Hamiltonian and Lagrangian systems
35Q55 NLS equations (nonlinear Schrödinger equations)
Full Text: DOI