Traveling and solitary wave solutions to the one-dimensional Gross-Pitaevskii equation

The evolution of traveling and solitary waves in Bose-Einstein condensates (BECs) with a time-dependent scattering length in an attractive/repulsive parabolic potential is studied. The homogeneous balance principle and the $F$-expansion technique are used to solve the one-dimensional Gross-Pitaevskii equation with time-varying coefficients. We obtained three classes of new exact traveling wave and localized solutions. Our results demonstrate that the BEC solitary wave solutions can be manipulated and controlled by the time-dependent scattering length.

Figure 1

(Color online) Bright breather solitary wave with the cosine scattering length. The parameters are: $\chi ={\chi }_0\text{cos}\left(t\right)$, ${\chi }_0=1$, $f_0=k_0=1$, $b_0=0$, $x_0=0$. Left: optical intensity distribution. Right: amplitude variation at the position $x=0$, as a function of time.

Figure 2

(Color online) Stable oscillations of a bright breather solitary wave, in the case $\chi =1+{\chi }_0\text{sin}\left(t\right)$, ${\chi }_0=0.1$, $f_0=k_0=b_0=1$, $x_0=0$. Left: amplitude of the wave. Right: velocity of the pulse center.

Figure 3

(Color online) Dynamics of the dark solitary wave compression, as given by Eq. (8). The parameters are given as follows: $\chi ={\chi }_0{e}^{2{\beta }_{0}t}$, ${\beta }_0=0.05$, $f_0=k_0=b_0={\chi }_0=1$, $x_0=0$. Left: intensity distribution as a function of $t$. Right: View from the above.

Figure 4

(Color online) Broadening evolution of a dark solitary wave, with the parameters: $\chi ={\chi }_0{e}^{-\alpha t}$, ${\chi }_0=1$, $f_0=k_0=1$, $x_0=0$. Left: intensity distribution for $\alpha =0.02$, $b_0=1$. Right: cross sections at $t=0,10,20$ from top to bottom, for $\alpha =0.04$, $b_0=0$.