The multi–phase averaging techniques and the modulation theory of the focussing and defocusing nonlinear schrodinger equations

Abstract

Multi–phase averaging techniques have been applied successfully in the investigations of the modulational and generalized Benjamine–Feir instabilities for the quasi–periodic, N–phase, inverse spectral solutions of KdV [1], sine–Gordon (s–G) [2,3,4], and focussing and defocusing nonlinear Schrodinger equation [5,10], The key is that the multi–phase averagings, as the N–fold integrals, can be transferred to the N–iterated integrals, and therefore, can be evaluated, which is essential in the analysis of PDE perturbations analyzed by the averaging methods. In this paper, the transformations from cerain N–fold integrals to the N–iterated integrals for NLS are developed rigorously, and made to be numerically computable. Those integrals are also closely related to KdV and s–G. As an application, the modulation theory of the modulating N–phuse NLS solutions are Presented, a result given by Forest and Lee in [5,10].