On the spectral problem associated with the time-periodic nonlinear Schr\"odinger equation

TL;DR

This paper advances the spectral analysis of the time-part of the Lax pair for the nonlinear Schrödinger equation with time-periodic solutions, aiming to develop an integrable systems framework similar to the space-periodic case.

Contribution

It performs a spectral analysis of the t-part of the Lax pair for time-periodic NLS solutions, extending the integrable systems approach.

Findings

Spectral analysis of the t-part for time-periodic solutions.

Development of a framework analogous to space-periodic case.

Progress towards a Birkhoff normal form for time-periodic NLS.

Abstract

According to its Lax pair formulation, the nonlinear Schr\"odinger (NLS) equation can be expressed as the compatibility condition of two linear ordinary differential equations with an analytic dependence on a complex parameter. The first of these equations---often referred to as the \emph{$x$-part} of the Lax pair---can be rewritten as an eigenvalue problem for a Zakharov-Shabat operator. The spectral analysis of this operator is crucial for the solution of the initial value problem for the NLS equation via inverse scattering techniques. For space-periodic solutions, this leads to the existence of a Birkhoff normal form, which beautifully exhibits the structure of NLS as an infinite-dimensional completely integrable system. In this paper, we take the crucial steps towards developing an analogous picture for time-periodic solutions by performing a spectral analysis of the \emph{$t$-part} of the Lax pair with a periodic potential.