A determining form for the damped driven Nonlinear Schr\"odinger Equation- Fourier modes case

TL;DR

This paper establishes a connection between the global attractor of the 1D damped, driven nonlinear Schrödinger equation and a finite-dimensional determining form, providing a new way to analyze long-term dynamics using Fourier modes.

Contribution

It introduces a determining form as an ODE in a trajectory space that captures the global attractor of the NLS, with improved estimates for the number of determining modes.

Findings

Global attractor embedded in the determining form dynamics

One-to-one correspondence between attractor trajectories and steady states

Enhanced estimate for the number of determining Fourier modes

Abstract

In this paper we show that the global attractor of the 1D damped, driven, nonlinear Schr\"odinger equation (NLS) is embedded in the long-time dynamics of a determining form. The determining form is an ordinary differential equation in a space of trajectories $X=C_b^1(\mathbb{R}, P_mH^2)$ where $P_m$ is the $L^2$-projector onto the span of the first $m$ Fourier modes. There is a one-to-one identification with the trajectories in the global attractor of the NLS and the steady states of the determining form. We also give an improved estimate for the number of the determining modes.