Quasiperiodicity and blowup in integrable subsystems of nonconservative nonlinear Schr\"odinger equations

Abstract

In this paper, we study the dynamics of a class of nonlinear Schr\"odinger equation $ i u_t = \triangle 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.