On the spectral problem associated with the time-periodic nonlinear Schrödinger equation

Abstract

According to its Lax pair formulation, the nonlinear Schrödinger (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 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 t-part of the Lax pair with a periodic potential.

The defocusing nonlinear Schrödinger equation with step-like oscillatory initial data

“…A rigorous Neumann series approach giving analytical meaning to these formal computations can be found e.g. in [37] for the large k behavior of mKdV eigenfunctions (of both Lax pair equations) to all orders; we refer to [41] for the leading order asymptotics of t-periodic NLS eigenfunctions.…”