We consider the nonlinear Schrödinger 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 , where α is a small constant. Assuming that the Neumann boundary value tends for large t to the periodic function , we show that can be expressed in terms of a perturbation series in α which can be constructed explicitly to any desired order. As an illustration, we compute to order α8 for the particular case that is the sum of two exponentials. We also show that there exist particular functions for which the above series can be summed up, and therefore, for these functions, can be obtained in closed form. The simplest such function is , where ω is a real constant.
- Received November 30, 2014.
- Accepted July 21, 2015.
- © 2015 The Author(s)
Published by the Royal Society. All rights reserved.