Manipulating matter rogue waves and breathers in Bose-Einstein condensates

We construct higher-order rogue wave solutions and breather profiles for the quasi-one-dimensional Gross-Pitaevskii equation with a time-dependent interatomic interaction and external trap through the similarity transformation technique. We consider three different forms of traps: (i) the time-independent expulsive trap, (ii) time-dependent monotonous trap, and (iii) time-dependent periodic trap. Our results show that when we change a parameter appearing in the time-independent or time-dependent trap the second- and third-order rogue waves transform into the first-order-like rogue waves. We also analyze the density profiles of breather solutions. Here we also show that the shapes of the breathers change when we tune the strength of the trap parameter. Our results may help to manage rogue waves experimentally in a BEC system.

Figure 1

Profiles of (a) soliton, (b) general breather, and (c) Peregrine soliton (rogue wave).

Figure 2

First, second, and third columns represent the formation of first-, second-, and third-order RWs in BECs for the time-dependent nonlinearity coefficient $R\left(t\right)=\text{sech}({\beta }_{0}t+\delta )$ and time-independent trap frequency $\beta {\left(t\right)}^2={\beta }_0^2$, obtained using expression (9). The parameter $r_0=0.5$ for panels (a), (e), and (i), $0.7$ for (b), (f), and (j), $0.9$ for (c), (g), and (k), and 1.2 for (d), (h), and (l). The other parameters are $c_1=0.01$, ${\beta }_0=0.1$, and $\delta =0.01$.

Figure 3

(a) First-order RW in BEC and (b) the corresponding contour plot obtained by numerically solving Eq. (1) through the split-step Crank-Nicolson method for the time-dependent nonlinearity coefficient $R\left(t\right)=\text{sech}({\beta }_{0}t+\delta )$ and time-independent trap frequency $\beta {\left(t\right)}^2={\beta }_0^2$. The initial condition chosen corresponds to the analytic solution of Fig. 2.

Figure 4

First-order RWs for $R\left(t\right)=\text{sech}({\beta }_{0}t+\delta )$ and $\beta {\left(t\right)}^2={\beta }_0^2$. The parameter ${\beta }_0$ is varied as (a) ${\beta }_0=0.1$, (b) ${\beta }_0=1.2$, (c) ${\beta }_0=5.0$. Panels (d)–(f) are their corresponding contour plots. The other parameters are fixed as $r_0=1.0$, $c_1=0.01$, and $\delta =0.01$.

Figure 5

Second-order RWs for $R\left(t\right)=\text{sech}({\beta }_{0}t+\delta )$ and $\beta {\left(t\right)}^2={\beta }_0^2$. The parameter ${\beta }_0$ is varied as (a)${\beta }_0=0.1$, (b) ${\beta }_0=1.2$, (c) ${\beta }_0=5.0$. Panels (d)–(f) are their corresponding contour plots. The other parameters are the same as in Fig. 4.

Figure 6

Third-order RWs for $R\left(t\right)=\text{sech}({\beta }_{0}t+\delta )$ and $\beta {\left(t\right)}^2={\beta }_0^2$. The parameter ${\beta }_0$ is varied as (a) ${\beta }_0=0.1$, (b) ${\beta }_0=1.5$, (c) ${\beta }_0=5.0$. Panels (d)–(f) are their corresponding contour plots. The other parameters are the same as in Fig. 4.

Figure 7

First-order RWs for $R\left(t\right)=1+tanh({\beta }_{0}t/2)$ and $\beta {\left(t\right)}^2={\beta }_0^2/2[1-tanh({\beta }_{0}t/2)]$. The parameter ${\beta }_0$ is varied as (a) ${\beta }_0=0.1$, (b) ${\beta }_0=1.0$, and (c) ${\beta }_0=5.0$. Panels (d)–(f) are the corresponding contour plots of (a)–(c). The other parameters are the same as in Fig. 4.

Figure 8

Second-order RWs for $R\left(t\right)=1+tanh({\beta }_{0}t/2)$ and $\beta {\left(t\right)}^2={\beta }_0^2/2[1-tanh({\beta }_{0}t/2)]$. The parameter ${\beta }_0$ is varied as (a) ${\beta }_0=0.1$, (b) ${\beta }_0=1.0$, and (c) ${\beta }_0=5.0$. Panels (d)–(f) are the corresponding contour plots of (a)–(c). The other parameters are the same as in Fig. 4.

Figure 9

Third-order RWs for $R\left(t\right)=1+tanh({\beta }_{0}t/2)$ and $\beta {\left(t\right)}^2={\beta }_0^2/2[1-tanh({\beta }_{0}t/2)]$. The parameter ${\beta }_0$ is varied as (a) ${\beta }_0=0.1$, (b) ${\beta }_0=1.0$, and (c) ${\beta }_0=5.0$. Panels (d)–(f) are their corresponding contour plots. The other parameters are the same as in Fig. 4.

Figure 10

(a) First-order RW, (b) second-order RW, and (c) third-order RW for $R\left(t\right)=1+cos\left(2{\beta }_{0}t\right)$ and $\beta {\left(t\right)}^2=2{\beta }_0^2[1+3{tan}^2\left({\beta }_{0}t\right)]$. Panels (d)–(f) are the corresponding contour plots of (a)–(c). The parameters are $r_0=1.0$, ${\beta }_0=2.5$, $c_1=0.01$, and $\delta =0.01$.

Figure 11

Triplet RWs. (a)–(c) $R\left(t\right)=\text{sech}({\beta }_{0}t+\delta )$ and $\beta {\left(t\right)}^2={\beta }_0^2$; (d)–(f) $R\left(t\right)=1+tanh({\beta }_{0}t/2)$ and $\beta {\left(t\right)}^2={\beta }_0^2/2[1-tanh({\beta }_{0}t/2)]$; (g)–(i) $R\left(t\right)=1+cos\left(2{\beta }_{0}t\right)$ and $\beta {\left(t\right)}^2=2{\beta }_0^2[1+3{tan}^2\left({\beta }_{0}t\right)]$. Further, in (a), (d), and (g) ${\beta }_0=0.1$, in (b), (e), and (h) ${\beta }_0=0.3$, and in (c), (f), and (i) ${\beta }_0=1.0$. The other parameters are $l=15$, $m=25$, $r_0=1.0$, $c_1=0.01$, and $\delta =0.01$.

Figure 12

Sextet RWs. (a)–(c) $R\left(t\right)=\text{sech}({\beta }_{0}t+\delta )$ and $\beta {\left(t\right)}^2={\beta }_0^2$; (d)–(f) $R\left(t\right)=1+tanh({\beta }_{0}t/2)$ and $\beta {\left(t\right)}^2={\beta }_0^2/2[1-tanh({\beta }_{0}t/2)]$; (g)–(i) $R\left(t\right)=1+cos\left(2{\beta }_{0}t\right)$ and $\beta {\left(t\right)}^2=2{\beta }_0^2[1+3{tan}^2\left({\beta }_{0}t\right)]$. Further, in (a), (d), and (g) ${\beta }_0=0.1$; in (b), (e), and (h) ${\beta }_0=0.25$; and in (c), (f), and (i) ${\beta }_0=1.0$. The other parameters are $l=10$, $m=20$, $g=500$, $h=500$, $r_0=1.0$, $c_1=0.01$, and $\delta =0.01$.

Figure 13

(a) AB for $\lambda =0.6i$, (b) MB for $\lambda =1.4i$ when $R\left(t\right)=\text{sech}({\beta }_{0}t+\delta )$ and $\beta {\left(t\right)}^2={\beta }_0^2$. (c) and (d) Their corresponding contour plots. The other parameters are $r_0=1.0$, ${\beta }_0=0.01$, $c_1=0.01$, and $\delta =0.01$.

Figure 14

Stretching of breathers. (a), (c), and (e) AB with an eigenvalue $\lambda =0.6i$; (b), (d), and (f) MB with $\lambda =1.4i$ for the three different forms of $R\left(t\right)$ discussed in the text. (a)–(d) ${\beta }_0=0.8$, (e) and (f) ${\beta }_0=1.5$. The other parameters are $r_0=1.0$, $c_1=0.01$, and $\delta =0.01$.

Figure 15

(a) Two AB profiles for ${\lambda }_1=0.55i$ and ${\lambda }_2=0.75i$ with $T_1=-3$ and $T_2=3$. (b) Two ABs without time shifts. (c) AB-RW profile for ${\lambda }_1=0.55i$ and ${\lambda }_2=0.99i$. (d) Two MBs for ${\lambda }_1=1.3i$ and ${\lambda }_2=1.4i$ with $X_1=-3$ and $X_2=3$. (e) Two MBs without space shifts and (f) the intersection of AB and MB for ${\lambda }_1=0.5i$ and ${\lambda }_2=1.3i$ for $R\left(t\right)=\text{sech}({\beta }_{0}t+\delta )$ and $\beta {\left(t\right)}^2={\beta }_0^2$. The other parameters are the same as in Fig. 4.

Figure 16

Distortion of two-breather profiles. (a), (d), and (g) Two ABs with ${\lambda }_1=0.55i$ and ${\lambda }_2=0.75i$; (b), (e), and (h) the intersection of AB-MB profiles with ${\lambda }_1=0.5i$ and ${\lambda }_2=1.3i$; and (c), (f), and (i) two MBs with ${\lambda }_1=1.3i$ and ${\lambda }_2=1.4i$. The trap parameter ${\beta }_0$ is chosen as (a) $0.15$, (b), and (e) $0.5$, (c) and (f) $0.2$, (d) $1.5$ and (g)–(i) $0.7$. The other parameters are $r_0=1.0$, $c_1=0.01$, and $\delta =0.01$.

Figure 17

Profiles: (a) Second-order RW and (b) third-order RW of the standard NLS equation.