Vector rogue waves and dark-bright boomeronic solitons in autonomous and nonautonomous settings

In this work we consider the dynamics of vector rogue waves and dark-bright solitons in two-component nonlinear Schrödinger equations with various physically motivated time-dependent nonlinearity coefficients, as well as spatiotemporally dependent potentials. A similarity transformation is utilized to convert the system into the integrable Manakov system and subsequently the vector rogue and dark-bright boomeronlike soliton solutions of the latter are converted back into ones of the original nonautonomous model. Using direct numerical simulations we find that, in most cases, the rogue wave formation is rapidly followed by a modulational instability that leads to the emergence of an expanding soliton train. Scenarios different than this generic phenomenology are also reported.

Figure 1

Exact rogue wave solutions in the absence (first and third rows) and presence (second and fourth rows) of self- and cross-coupling parameters. Panels (a) and (b) and (c) and (d) correspond to values of the parameters of $a_1=0.8,a_2=0,f=0,\sigma =1$, and $\alpha =0.05$, whereas (e) and (f) and (g) and (h) correspond to values of the parameters of $a_1=0.6,a_2=0.8,f=0,\sigma =0.5$, and $\alpha =1.8$.

Figure 2

Contour plots of the density profiles $|E_1{(t,z)|}^2$ (left panel) and $|E_2{(t,z)|}^2$ (right panel) over space-time corresponding to the exact solutions in the form of boomeronic DB solitons in the presence of self- and cross-coupling parameters with $a_1=0,a_2=1.3,f=0.15,\sigma =0.8$, and $\alpha =0.1$.

Figure 3

Typical form of variable nonlinearity coefficient $\gamma \left(z\right)$ for $\varepsilon =0.75$ and trap frequency $F\left(z\right)$ for various values of $\varepsilon$.

Figure 4

Profiles of nonlinearity $\gamma \left(z\right)$ (for $\varepsilon =0.55$) and trap frequency $F\left(z\right)$ given by Eqs. (11a) and (11b), respectively.

Figure 5

Dynamics of rogue waves with a periodically modulated nonlinearity $\gamma \left(z\right)$ given by Eq. (10a) and the corresponding trap frequency of Eq. (10b). Left and right panels correspond to density profiles of the numerically obtained fields $E_1$ and $E_2$, respectively. The top panels (a) and (b) show the spatial distribution of the intensities $|E_j{|}^2 \left(j=1,2\right)$ evaluated at $z=0$, whereas the second row panels (c) and (d) show their corresponding temporal evolution at $t=0$. The third row panels (e) and (f) show contour plots of the density profiles of the corresponding rogue waves and the fourth row ones (g) and (h) zoom-ins of (e) and (f).

Figure 6

Dynamics of coexisting Peregrine and DB boomeronic solitons with a periodically modulated nonlinearity (10a). Left and right panels correspond to density profiles of the numerically obtained fields $E_1$ and $E_2$, respectively. The top row panels (a) and (b) show the spatial distribution of the intensities $|E_1{|}^2$ at $z=0$ (left) and $|E_2{|}^2$ at $z=0.19$ (right), whereas the second row panels (c) and (d) show their corresponding temporal evolution at $t=-0.1$ and $t=0$, respectively. Again, the panels (e)–(h) show contour plots of the density profiles of the corresponding waves.

Figure 7

Same as Fig. 5 but for the kinklike modulated nonlinearity given by Eq. (11a) and the trap frequency of Eq. (11b). Left and right panels correspond to density profiles of the numerically obtained fields $E_1$ and $E_2$, respectively. The top row panels (a) and (b) show the spatial distribution of the intensities $|E_j{|}^2 \left(j=1,2\right)$ evaluated at $z=0.01$, whereas the second row panels (c) and (d) show their corresponding temporal evolution at $t=0$. The panels (e)–(h) show contour plots of the density profiles of the corresponding rogue waves.

Figure 8

Same as Fig. 6 but for the kinklike modulated nonlinearity given by Eq. (11a). Left and right panels correspond to density profiles of the numerically obtained fields $E_1$ and $E_2$, respectively. The top row panels (a) and (b) show the spatial distribution of the intensities $|E_1{|}^2$ at $z=0.01$ (left) and $|E_2{|}^2$ at $z=0.02$ (right), whereas the second row panels (c) and (d) show their corresponding temporal evolution at $t=0$. The panels (e)–(h) show contour plots of the density profiles of the corresponding rogue waves.