High-order rogue wave and mixed interaction patterns for the three-component Gross-Pitaevskii equations in $F=1$ spinor Bose-Einstein condensates

Under investigation are the three-component Gross-Pitaevskii equations in $F=1$ spinor Bose-Einstein condensates. Various localized waves' generation mechanisms have been derived from plane wave solutions using modulation instability. The perturbed continuous waves produce a large number of rogue wave structures through the split-step Fourier numerical method. Based on the known Lax pair, we construct the generalized iterative $(n,N-n)$-fold Darboux transformation to generate various high-order solutions, including the bright-dark-bright structure of rogue waves, periodic waves, and their mixed interaction structures. Numerical simulations show that rogue waves with a two-peaked structure have robust noise resistance and stable dynamical behavior. The asymptotic states of high-order rogue waves as the parameter approaches infinity are also predicted using the large parameter asymptotic technique. In addition, the position of these localized wave patterns can be controlled by some special parameters. These results may help us understand the dynamic behavior of spinor condensates for the mean-field approximation.

Figure 1

(a) The MI growth rate $G\left(\kappa \right)$ with $a=0$. (b) MI gain spectrum distributed with respect to $c_i(i=1,2)$, $c_j=1(j\ne 0)$, and $\kappa$ in the range $[0,10]\times [0,20]$. The yellow dotted line separates the MI and MS regions. The white dotted line represents the curve $C$. Red crosses denote maximum gain points.

Figure 2

Numerical evolution of $|{\mathrm{\Phi }}_m|(m=\pm 1,0)$ from a plane wave background with the Gaussian pulse perturbation ${\mathrm{\Phi }}_m^{\left[0\right]}(0,t)(1+0.1e^{-t^2})$. (a1) Numerical evolution of $|{\mathrm{\Phi }}_{+1}|$. (a2) Numerical evolution of $|{\mathrm{\Phi }}_0|$. (a3) Numerical evolution of $|{\mathrm{\Phi }}_{-1}|$.

Figure 3

The first-order RW structures with $c=1$ and ${\lambda }_1=\sqrt{2}$. (a1)–(c1) $e_0=d_0=0$. (a2)–(c2) $e_0=0.5$ and $d_0=0$.

Figure 4

(a1)–(a3) Numerical evolution of $|{\mathrm{\Phi }}_m|(m=\pm 1,0)$ from the plane wave background ${\mathrm{\Phi }}_m^{\left[0\right]}(0,t)$. (b1)–(b3) Numerical evolution of ${\mathrm{\Phi }}_m(m=\pm 1,0)$ from the plane wave background with small noise.

Figure 5

Density profiles of first-order RW solutions at the moment $t=-e_0$. (a) Local density of the first-order RW for each component. (b) Particle density $\overline{n}$ . (c) Spin density ${(f_x,f_y,f_z)}^T$.

Figure 6

The first-order PW structures with $c=1$ and ${\lambda }_1=\sqrt{2}+1$. (a1)–(c1) $e_0=0$ and $d_0=-0.1$. (a2)–(c2) $e_0=d_0=-0.1$.

Figure 7

Second-order RW solutions with parameters $c=1$, ${\gamma }_2=0$, ${\gamma }_3=2$, and $h={}^3\sqrt{75}/2$. Panels (a)–(c) represent each of the three components of the RW solutions, with the left-hand side of each panel representing the locations of the three first-order RWs.

Figure 8

The RW patterns (range from $N=1$ to $N=5$) for ${\mathrm{\Phi }}_{+1}$ with $c=1$ and ${\lambda }_1=\sqrt{2}$. The different selection of separation parameters, all of which are shown in Table 1, makes them present various structures.

Figure 9

The RW patterns (range from $N=1$ to $N=5$) for ${\mathrm{\Phi }}_0$ with $c=1$ and ${\lambda }_1=\sqrt{2}$. The remaining parameters are chosen to be the same as those in Fig. 8.

Figure 10

The RW patterns (range from $N=1$ to $N=5$) for ${\mathrm{\Phi }}_{-1}$ with $c=1$ and ${\lambda }_1=\sqrt{2}$. The remaining parameters are chosen to be the same as those in Fig. 8.

Figure 11

The mixed interaction patterns (range form $N=2$ to $N=4$) for ${\mathrm{\Phi }}_{+1}$ with $c=1$, ${\lambda }_1=\sqrt{2}$, and ${\lambda }_2=\sqrt{2}+1$. The parameters $e$ (see the top of the four diagrams on the right) makes the four diagrams on the left have separated structures.

Figure 12

Density plots corresponding to each panel in Fig. 11.

Figure 13

The mixed interaction patterns (range form $N=2$ to $N=4$) for ${\mathrm{\Phi }}_0$ with $c=1$, ${\lambda }_1=\sqrt{2}$, and ${\lambda }_2=\sqrt{2}+1$.

Figure 14

Density plots corresponding to each panel in Fig. 13.

Figure 15

The mixed interaction patterns (range form $N=2$ to $N=4$) for ${\mathrm{\Phi }}_{-1}$ with $c=1$, ${\lambda }_1=\sqrt{2}$, and ${\lambda }_2=\sqrt{2}+1$.

Figure 16

Density plots corresponding to each panel in Fig. 15.