Solutions Of Three-Dimensional Nonlinear Schrdinger Equation Using Green Function Method Applied To Gravity Waves

Abstract

In this paper, we investigate the time independent Schrdinger equation which has complex valued potential function under the general point interaction. We construct Green function of this problem and we nd the resolvent of the problem in terms of Green function.We study the effects of multiple scattering of slowly modulated water waves by a weakly random bathymetry. The combined effects of weak nonlinearity, dispersion and random irregularities are treated together to yield a nonlinear Schrodinger equation applied to gravity wave with a complex damping term. Additionally based on the small-signal analysis theory, the nonlinear Schrodinger equation (NLSE) with fiber loss over gravity wave is solved. It is also adapted to the NLSE with the high-order dispersion terms. Furthermore, a general theory on cross-phase modu- lation (XPM) intensity fluctuation which adapted to all kinds of modulation for- mats (continuous wave, non-return-to-zero wave, and return-zero pulse wave) is presented. Secondly, by the Green function method, the NLSE is directly solved in the time domain. It does not bring any spurious effect compared with the split-step method in which the step size has to be carefully controlled. Finally, the fourth-order dispersion coefficient of fibers can be estimated by the Green function solution of NLSE tha applied to gravity wave. The fourth-order dispersion coefficient varies with distance slightly and is about 0:002ps=km, 0:003ps=nm, and 0:00032 ps=nm for SMF, NZDSF, and DCF, respectively. In the zero-dispersion regime, the higher-order nonlinear effect (higher than self-steepening) has a strong impact on the short gravity wave shape, but this effect degrades rapidly with the increase of b2.