2014/12/31 by J. Lenells, A. S. Fokas · 1 citation
Mathematics · Physics and Astronomy · #math.AP #nlin.SI #msc:35Q55 #msc:37K15
paper · pdf · doi:10.1098/rspa.2014.0926
25 pages
arxiv created 2015/09/19 · arxiv updated 2016/02/17
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 αg0b(t), where α is a small constant. Assuming that the Neumann boundary value tends for large t to the periodic function g1b(t), we show that g1b(t) 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 g1b(t) to order α8 for the particular case that g0b(t) is the sum of two exponentials. We also show that there exist particular functions g0b(t) for which the above series can be summed up, and therefore for these functions g1b(t) can be obtained in closed form. The simplest such function is exp(iωt), where ω is a real constant.