A Perturbative Approach for the Asymptotic Evaluation of the Neumann Value Corresponding to the Dirichlet Datum of a Single Periodic Exponential for the NLS

Abstract

Boundary value problems for the nonlinear Schrödinger equation formulated on the half-line can be analyzed by the Fokas method. For the Dirichlet problem, the most difficult step of this method is the characterization of the unknown Neumann boundary value. For the case that the Dirichlet datum consists of a single periodic exponential, namely, a exp(iωt), a, ω real, it has been shown in [2–4] that if one assumes that the Neumann boundary value is given for large t by c exp(iωt), then c can be computed explicitly in terms of a and ω. Here, using the perturbative approach introduced in [16], it is shown that for typical initial conditions, it is indeed the case that at least up to third order in a perturbative expansion the Neumann boundary value is given by c exp(iωt) and the value of c is at least up to this order the value found in [2–4].