Modulation instability induced by four coupled of matter waves with two-and three-body interactions

Abstract

Modulation instabilities of matter waves described by a system of four coupled Gross–Pitaevskii equations with two-and three-body interactions are analytically and numerically investigated. For analytical treatment, we use the linear stability analysis to derive the dispersion relation which allows us to predict regions of modulation instabilities gain spectra in two regimes. In the first regime, two counter-propagating beams of matter waves intensities are different, while in the second, their intensities are equal. It is shown that both the two-and three-body interactions greatly affect the gain spectra in the two regimes. By full numerical simulations of four coupled Gross–Pitaevskii equations, we confirm the analytical predictions. Moreover, we explore with the same parameters two possibilities of modulation instabilities for different velocities v that depend on the two-body interactions. For the small values of two-body interactions, when v > 1, the spatiotemporal evolution appears to be stable for long times, the modulation instability is enhanced. On the contrary, when v < 1, this evolution becomes unstable, the modulation instability tends to be suppressed.

Effects of residual nonlinearities on the modulational instability of three-component Bose–Einstein condensates

“…Hence we set ( ) V x 0 = . Then, we seek the solutions to the governing equations (8)- (10) in the form of plane waves as below.…”

“…where kʼs and μʼs are the wave-number and frequency of the continuous wave background for each of the three components. The dispersion relation μ(k) can be obtained by inserting the plane wave ansatz into equations (8)- (10).…”

“…Bose-Einstein condensates [1][2][3] are useful systems for studying modulational instability, solitons and nonlinear matter-wave dynamics [4][5][6][7][8][9][10] and rogue waves [11][12][13]. The modulation instability causes a uniform medium to break up into pulsing solitary waves.…”