Spatiotemporal wave and soliton solutions to the generalized $\left(3+1\right)$-dimensional Gross-Pitaevskii equation

Exact extended traveling wave and spatiotemporal soliton solutions to the generalized $\left(3+1\right)$-dimensional Gross-Pitaevskii equation with time-dependent coefficients are obtained. The case with constant diffraction and parabolic potential strength, but with variable gain, is discussed in some detail. It is found that gain in the system is necessary for the appearance of stable solitons.

Figure 1

(Color online) Decaying bent soliton solutions to GPE as functions of time, for $b_0=1$. Intensity ${\left|u\right|}^2$ for (a) $F=\text{tanh}$ and (b) $F=\text{sech}$ presented as functions of $k_{0}x+l_{0}y+m_{0}z$ and $t$. Coefficients: $\beta =1$, $\alpha =1$, $\gamma \left(t\right)=-0.05$, $a_0=0$, $e_0=0$, $k_0=l_0=m_0=1$, ${\omega }_0=0$, and $ϵ=0$.

Figure 2

(Color online) Decaying straight soliton solutions to GPE as functions of time. The setup and parameters are the same as in Fig. 1 except for $b_0=0$.

Figure 3

(Color online) Decaying traveling wave solutions, given in terms of JEFs. The setup and parameters are the same as in Fig. 2, except for $M=0.99$. (a) $F=\text{sn}$ and (b) $F=\text{cn}$.

Figure 4

(Color online) Bent soliton solutions as functions of time. (a) Dark; (b) Bright. The setup and parameters are the same as in Fig. 1, except for $\gamma \left(t\right)=3/\sqrt{2}$, the critical value of $\gamma$.

Figure 5

(Color online) Straight soliton solutions as functions of time. The parameters are the same as in Fig. 4 except for $b_0=0$.

Figure 6

(Color online) Traveling wave solutions in terms of JEFs. The parameters are the same as in Fig. 5, except for $M=0.99$. (a) $F=\text{sn}$ and (b) $F=\text{cn}$.