The nonlinear Schr\"odinger equation with $t$-periodic data: II. Perturbative results

TL;DR

This paper develops a perturbative approach to analyze the nonlinear Schr"odinger equation on the half-line with periodic boundary data, providing explicit series solutions and identifying cases where the series can be summed to closed form.

Contribution

It introduces a perturbation series method for the boundary value problem with periodic data and explicitly computes high-order terms, including cases with closed-form solutions.

Findings

Perturbation series for the Neumann boundary value can be constructed explicitly.

Explicit computation of the series up to order  for specific boundary functions.

Identification of functions allowing the series to be summed to closed form.

Abstract

We consider the nonlinear Schr\"odinger equation on the half-line with a given Dirichlet boundary datum which for large $t$ tends to a periodic function. We assume that this function is sufficiently small, namely that it can be expressed in the form $\alpha g_0^b(t)$, where $\alpha$ is a small constant. Assuming that the Neumann boundary value tends for large $t$ to the periodic function $g_1^b(t)$, we show that $g_1^b(t)$ can be expressed in terms of a perturbation series in $\alpha$ which can be constructed explicitly to any desired order. As an illustration, we compute $g_1^b(t)$ to order $\alpha^8$ for the particular case that $g_0^b(t)$ is the sum of two exponentials. We also show that there exist particular functions $g_0^b(t)$ for which the above series can be summed up, and therefore for these functions $g_1^b(t)$ can be obtained in closed form. The simplest such function is $\exp(i\omega t)$, where $\omega$ is a real constant.