Quasiperiodicity and blowup in integrable subsystems of nonconservative nonlinear Schrödinger equations

Abstract

In this paper, we study the dynamics of a class of nonlinear Schrödinger equation $i{u}_t=△u+u^p$ for $x\in {\mathbb{T}}^d$. We prove that the PDE is integrable on the space of non-negative Fourier coefficients, in particular that each Fourier coefficient of a solution can be explicitly solved by quadrature. Within this subspace we demonstrate a large class of (quasi)periodic solutions all with the same frequency, as well as solutions which blowup in finite time in the $L^2$ norm.

24 pages, 1 figure