Unidirectional segregation of bright-bright soliton through a parity-time-symmetric potential

We study the dynamics of two-component vector solitons, namely, bright-bright (BB) solitons interacting with parity-time ($\mathcal{PT}$)-symmetric potentials. We employ direct numerical simulations to demonstrate the unidirectional segregation of the BB soliton. Using a modified perturbed dynamical variational Lagrangian approximation, we develop an analytical model to verify the results obtained from numerical calculations. Simplified variational equations of motion suggest that the splitting of BB solitons can be explained by considering the effective force between the two components.

Figure 1

Density plot of a two-component BB soliton (a) interacting with a $\mathcal{PT}$-symmetric potential (b). The BB soliton parameters are ${\xi }_1=-18, {\xi }_2=-21, v_1=v_2=v_0=0.24$. The potential parameters are $V_0=4, W_0=2$. Here, we set $\beta =-1$ in Eq. (3).

Figure 2

Density plot of two-component BB soliton (a) scattering by a $\mathcal{PT}$-symmetric potential, (b). We obtain unidirectional segregation using the same parameters as in Fig. 1 but with $\beta =1$.

Figure 3

Transport coefficients obtained from the numerical solution of the coupled NLSEs, Eq. (1), using the potential, Eq. (3) with $\beta =-1$. (a) [(b)] represents the first (second) component $u(x,t) \left[v\right(x,t\left)\right]$. For the velocity range, both components transmitted over the potential.

Figure 4

Transport coefficients obtained from the numerical solution of the coupled NLSEs, Eq. (1), using the potential, Eq. (3) with $\beta =1$. (a) [(b)] represents the first (second) component $u(x,t) \left[v\right(x,t\left)\right]$. Depending on the incident velocity, we have a total reflection of both components, unidirectional segregation, or total transmission.

Figure 5

Density plot of a two-component BB soliton scattering by a $\mathcal{PT}$-symmetric potential, Eq. (4) with $\beta =-1$. The solid lines represent the numerical calculations for the two components, whereas the dashed lines represent the variational calculations. The potential parameters are $V_0=-0.1, W_0=-0.08$. The locations of the two components are ${\xi }_1=-20, {\xi }_2=-24$ with $v_0=0.33$. The coupling strength, $g=0.05$.

Figure 6

Density plot of a two-component BB soliton scattering by a $\mathcal{PT}$-symmetric potential, Eq. (4) with $\beta =1$. The solid lines represent the numerical calculations for the two components, whereas the dashed lines represent the variational calculations. The potential and BB soliton parameters are similar to Fig. 5.

Figure 7

Transport coefficients obtained from the numerical solution of the coupled NLSEs, Eq. (1), using Eq. (4) with $\beta =1$. (a) [(b)] represents the first (second) component $u(x,t) \left[v\right(x,t\left)\right]$. For the velocity range, both components transmitted over the potential.

Figure 8

Transport coefficients obtained from the numerical solution of the coupled NLSEs, Eq. (1), using Eq. (4) with $\beta =-1$. (a) [(b)] represents the first (second) component $u(x,t) \left[v\right(x,t\left)\right]$. Depending on the incident velocity, we have a total reflection of both components, unidirectional segregation, or total transmission.

Figure 9

Transport coefficients obtained from the variational calculations using Eq. (4) with $\beta =1$. (a) [(b)] represents the first (second) component $u(x,t) \left[v\right(x,t\left)\right]$. For the velocity range, both components transmitted over the potential.

Figure 10

Transport coefficients obtained from the variational calculations using Eq. (4) with $\beta =-1$. (a) [(b)] represents the first (second) component $u(x,t) \left[v\right(x,t\left)\right]$. Depending on the incident velocity, we have a total reflection of both components, unidirectional segregation, or total transmission.

Figure 11

Effective force $-f_1\left[{\xi }_1\right]$ for different coupling strengths $g$. Parameters are as follows: $V_0=-0.1, W_0=-0.08, a=1, {\xi }_2=0, N=2, c_1=0$, and $L=0.5$.