The nonlinear Schrödinger equation with t -periodic data: II. Perturbative results

Abstract

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 α g 0 b ( t ) , where α 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 α which can be constructed explicitly to any desired order. As an illustration, we compute g 1 b ( t ) to order α 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 ω t ) , where ω is a real constant.