Quasi-Töplitz functions in KAM theorem (Q2855123)

The nonlinear Schrödinger equation (NLS) is considered on the torus in the space of \(d\) dimensions. The authors prove existence and stability of quasi-periodic solutions in this model, which is constructed from the zero equilibrium by continuation arguments. The linearization at the zero solution admits a quasi-periodic solution with \(b\) frequencies postulated from the linear term of the nonlinear Schrödinger equation.NEWLINENEWLINEThe authors use the standard technique based on the KAM theorem. However, two new modifications are made compared to the previous literature, in particular, compared with the recent work of \textit{L. H. Eliasson} and \textit{S. B. Kuksin} [Ann. Math. (2) 172, No. 1, 371--435 (2010; Zbl 1201.35177)]. First, the authors use the fact that the NLS equation has the total momentum as an integral of motion. Second, properties of the quasi-Töplitz functions are used. Not only significant simplifications are achieved by using these new elements, but the bifurcating solutions are also proved to be analytic in the space and time variables.