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

TL;DR

This paper investigates a class of nonlinear Schr"odinger equations on a torus, demonstrating integrability in a specific subspace, and identifying both quasiperiodic solutions and finite-time blowup phenomena.

Contribution

It proves integrability of the PDE on a Fourier coefficient subspace and constructs explicit quasiperiodic and blowup solutions.

Findings

Fourier coefficients can be explicitly solved by quadrature.

Existence of large classes of quasiperiodic solutions.

Identification of finite-time blowup solutions.

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.