Collapse and evolution of wave field based on a generalized nonlinear Schrdinger equation

Abstract

A generalized nonlinear Schrdinger equation is numerically studied using the split-step Fourier method. For a fixed external potential field and an initial pulse disturbance, the effects of the complex coefficients p and q in the nonlinear Schrdinger equation on the evolution of the wave field are investigated. From a large number of simulations, it is found that the evolution of the wave field remains similar for different signs of the real parts of p and q, and different values of the real part of p. The initial pulse consisting of the longest wavelength modes (in the smallest-|k| corner of the phase space) of the spectrum first suffers modulational instability. Collapse begins at t～0.1, followed by inverse cascade of the shortest wavelength modes to longer wavelength ones, so that the whole k space becomes turbulent. For p = 1+0.04i, and q = 1+0.6i, it is found that first modulational instability occurs in the longer wavelength regime and the wave energy is transferred to the larger |k| modes. Then the wave collapse appears with increasing wave energy. Next, the large-|k| modes condense into a smaller-|k| mode by inverse cascade before spreading to the center of the phase space, until a turbulent state occurs there. Finally, most of the wave energy is condensed to the neighborhoods of three modes.