The multi-phase averaging techniques and the modulation theory of the focussing and defocusing nonlinear schrodinger equations
Title:
The multi-phase averaging techniques and the modulation theory of the focussing and defocusing nonlinear schrodinger equations
Author:
Lee, Jong-Eao
Appeared in:
Communications in partial differential equations
Paging:
Volume 15 (1990) nr. 9 pages 1293-1311
Year:
1990
Contents:
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].
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