ODINGER EQUATION WITH t-PERIODIC DATA: II. PERTURBATIVE RESULTS

We consider the nonlinear Schrodinger 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 formg b(t), whereis a small constant. Assuming that the Neumann boundary value tends for large t to the periodic function g b(t), we show that g b(t) can be expressed in terms of a perturbation series inwhich can be constructed explicitly to any desired order. As an illustration, we compute g b(t) to order � 8 for the particular case that g b(t) is the sum of two exponentials. We also show that there exist particular functions g b(t) for which the above series can be summed up, and therefore for these functions g b(t) can be obtained in closed form. The simplest such function is exp(i!t), where ! is a real constant.