Darboux Transformation for the 3-Dimension Nonlinear Schrodinger Equation

The transient nonlinear Schrodinger equation (NLSE) is solved with the Darboux transformation. Based on this solution, the longitude and transverse field evolution is presented; the time-varying coefficients (dispersion and nonlinear parameters) of NLSE are determined; the dynamical properties of the photonic crystal fiber-based metamaterials and the turning from loss to gain in photonic crystal fibers are interpreted; the resonance effect is redefined; and the resonance frequency which is a function of field is calculated.