Supercritical nonlinear Schrödinger equations: quasi-periodic solutions and almost global existence (Q2837191)

The paper addresses a multidimensional nonlinear Schrödinger equation with a general self-focusing nonlinearity, NEWLINE\[NEWLINEiu_t = -\Delta u+ |u|^{2p}u + \dots,NEWLINE\]NEWLINE to which additional terms may be added. Here \(p\geq 1\) is an integer, and the equation is posed in the space of arbitrary dimension \(D\), with periodic boundary conditions in all directions, i.e., on a \(D\)-dimensional torus. The periods in all directions are \(2\pi\). The linearized version of this equation can be solved, in an obvious way, by means of the Fourier transform, and a generic solution may be quasiperiodic in time, represented by a mixture of several different frequencies. The main result is an almost global existence of time-quasiperiodic solutions in the full nonlinear equation. The proof is carried out by means of a method of geometric selection in the Fourier space.