Fourier mode dynamics for the nonlinear Schroedinger equation in one-dimensional bounded domains

TL;DR

This paper investigates the dynamics of the 1D nonlinear Schrödinger equation in bounded domains, revealing collapse and slow Fourier mode cascades, with insights into boundary effects and frequency filtering to prevent collapse.

Contribution

It introduces a simplified amplitude equation framework for resonant Fourier modes, explaining slow dynamics and the role of the zero mode under different boundary conditions.

Findings

Collapse occurs rapidly in the focusing nonlinear Schrödinger equation.

Slow Fourier mode cascades are governed by amplitude equations for resonant modes.

Filtering high frequencies can prevent collapse by controlling mode interactions.

Abstract

We analyze the 1D focusing nonlinear Schr\"{o}dinger equation in a finite interval with homogeneous Dirichlet or Neumann boundary conditions. There are two main dynamics, the collapse which is very fast and a slow cascade of Fourier modes. For the cubic nonlinearity the calculations show no long term energy exchange between Fourier modes as opposed to higher nonlinearities. This slow dynamics is explained by fairly simple amplitude equations for the resonant Fourier modes. Their solutions are well behaved so filtering high frequencies prevents collapse. Finally these equations elucidate the unique role of the zero mode for the Neumann boundary conditions.