Nonautonomous discrete bright soliton solutions and interaction management for the Ablowitz-Ladik equation

We present the nonautonomous discrete bright soliton solutions and their interactions in the discrete Ablowitz-Ladik (DAL) equation with variable coefficients, which possesses complicated wave propagation in time and differs from the usual bright soliton waves. The differential-difference similarity transformation allows us to relate the discrete bright soliton solutions of the inhomogeneous DAL equation to the solutions of the homogeneous DAL equation. Propagation and interaction behaviors of the nonautonomous discrete solitons are analyzed through the one- and two-soliton solutions. We study the discrete snaking behaviors, parabolic behaviors, and interaction behaviors of the discrete solitons. In addition, the interaction management with free functions and dynamic behaviors of these solutions is investigated analytically, which have certain applications in electrical and optical systems.

Figure 1

Profiles of (a) nonlinearity $v_1\left(t\right)$ is given by Eq. (9) with $a_1\left(t\right)=\text{sin}\left(t\right),a_2\left(t\right)=\text{cos}\left(3t\right)$, $a=1,b=1$, $h=1$; (b) $g\left(t\right)$ is given by Eq. (8) with $\gamma \left(t\right)=\text{sin}\left(t\right)\text{sin}\left(2t\right),{\rho }_0=1$; (c) the function $\tau \left(t\right)$ is given by Eq. (7) with $a_1\left(t\right)=sin\left(3t\right),a_2\left(t\right)=0,a=1,b=1$.

Figure 2

Profiles of (a) the intensity distribution $|{\Psi }_n^{\left(1\right)}\left(t\right)|$ with $k=0.8,h=1,a=1,b=1,\gamma \left(t\right)=-6\text{sin}\left(t\right){\text{cos}}^5\left(t\right),{\rho }_0=1$; (b) the intensity distribution $|{\Psi }_n^{\left(1\right)}\left(t\right)|$ for $t=0$ [the blue (left) line], $t=0.5$ [the red (middle) line], $t=2$ [the green (right) line]; (c) the intensity distribution $|{\Psi }_n^{\left(1\right)}\left(t\right)|$ with the parameters as $k=1,h=1,a=1,b=1,a_1\left(t\right)=a_2\left(t\right)=\frac{1-2t}{\sqrt{2}},\gamma \left(t\right)=-\text{sin}\left(t\right)\text{cos}\left(t\right),{\rho }_0=1$ in Eq. (10).

Figure 3

Profiles of (a) the intensity distribution $|{\Psi }_n^{\left(1\right)}\left(t\right)|$ with $k=1.5+i,h=1,a=1,b=0,a_1\left(t\right)=0.01\text{sin}\left(t\right),\gamma \left(t\right)=\text{sin}\left(t\right),{\rho }_0=0.1$; (b) the intensity distribution $|{\Psi }_n^{\left(1\right)}\left(t\right)|$ for $t=0$ [the blue (upper) line], $t=1$ [the red (middle) line], $t=2$ [the green (lower) line]; (c) the intensity distribution $|{\Psi }_n^{\left(1\right)}\left(t\right)|$ with $k=0.8,h=1,a=1,b=0,a_1\left(t\right)=0.15-0.01t^2,\gamma \left(t\right)=-0.1\text{cos}\left(t\right),{\rho }_0=1$ in Eq. (11).

Figure 4

Profile of (a) the intensity distribution $|{\Psi }_n^{\left(2\right)}\left(t\right)|$ with $k_1=2+0.6i,k_2=1.5+0.8i,h=1.2,a=\sqrt{2}/2,b=\sqrt{2}/2,a_1\left(t\right)=\text{sin}\left(t\right),a_2\left(t\right)=\text{cos}\left(t\right),\gamma \left(t\right)=0,{\rho }_0=1$; (b) the intensity distribution $|{\Psi }_n^{\left(2\right)}\left(t\right)|$ with $k_1=2+0.6i,k_2=1.5+0.8i,h=1.2,a=1,b=1,a_1\left(t\right)=0.1{\text{sin}}^2\left(t\right),a_2\left(t\right)=0.1{\text{cos}}^2\left(t\right),\gamma \left(t\right)=0.2\text{sin}\left(t\right),{\rho }_0=1$; (c) the intensity distribution $|{\Psi }_n^{\left(2\right)}\left(t\right)|$ for $k_1=2+0.6i,k_2=1.5+0.8i,h=1.2,a=1,b=1,a_1\left(t\right)=0.5\text{sin}\left(t\right),a_2\left(t\right)=0.5\text{cos}\left(t\right),\gamma \left(t\right)=0.2\text{sin}\left(t\right)\text{cos}\left(t\right),{\rho }_0=1$ in Eq. (12).

Figure 5

Profiles of (a) the intensity distribution $|{\Psi }_n^{\left(2\right)}\left(t\right)|$ with $k_1=2+0.6i,k_2=1.5+0.8i,h=1.2,a_1\left(t\right)=1,\gamma \left(t\right)=0.1\text{sin}\left(t\right)$; (b) the intensity distribution $|{\Psi }_n^{\left(2\right)}\left(t\right)|$ with $k_1=2+0.6i,k_2=1.5+0.8i,h=1.2,a_1\left(t\right)=\text{cos}\left(t\right),\gamma \left(t\right)=\text{sin}\left(t\right)$; (c) the intensity distribution $|{\Psi }_n^{\left(2\right)}\left(t\right)|$ with $k_1=2+0.6i,k_2=1.5+0.8i,h=1.2,a_1\left(t\right)=\text{sn}(3t,1),\gamma \left(t\right)=0.1\text{cos}\left(t\right)$; (d) the intensity distribution $|{\Psi }_n^{\left(2\right)}\left(t\right)|$ with $k_1=2+0.6i,k_2=1.5+0.8i,h=1.2,a_1\left(t\right)=\text{sn}(3t,2),\gamma \left(t\right)=0.1\text{cos}\left(t\right)$ in Eq. (1).