Solitons in one-dimensional nonlinear Schrödinger lattices with a local inhomogeneity

In this paper we analyze the existence, stability, dynamical formation, and mobility properties of localized solutions in a one-dimensional system described by the discrete nonlinear Schrödinger equation with a linear point defect. We consider both attractive and repulsive defects in a focusing lattice. Among our main findings are (a) the destabilization of the on-site mode centered at the defect in the repulsive case, (b) the disappearance of localized modes in the vicinity of the defect due to saddle-node bifurcations for sufficiently strong defects of either type, (c) the decrease of the amplitude formation threshold for attractive and its increase for repulsive defects, and (d) the detailed elucidation as a function of initial speed and defect strength of the different regimes (trapping, trapping and reflection, pure reflection, and pure transmission) of interaction of a moving localized mode with the defect.

Figure 1

Linear modes: (a) The dispersion relation as a function of impurity parameter $\alpha$ (notice the impurity mode outside of the interval $[-2C,2C]$). The linear modes are normalized $({\sum }_n\mid {\varphi }_n{\mid }^2=1)$, the impurity is located at $n=0$, and periodic boundary conditions are considered. The other panels depict examples of the profiles of the impurity modes. (b) Profile for $\alpha =-1$ (repulsive impurity), and (c) profile for $\alpha =1$ (attractive impurity). In all cases $N=200$ and $C=1$.

Figure 2

Bifurcation diagram of stable (solid line) and unstable (dashed line) nonlinear modes. Shown is the power $P$ as a function of the impurity parameter $\alpha$. In all cases $N=100$ and $\omega =2.5$. The panel (a) is for $C=1.0$, while the panel (b) is for $C=0.2$. The branch designation is as follows: $A$ unstable soliton centered at the impurity $(n=n_0)$, $B$ stable on-site soliton centered at $n=n_0$, $C$ unstable intersite soliton centered at $n=n_0+0.5$, $D$ stable on-site soliton at $n=n_0+1$, $E$ unstable intersite soliton at $n=n_0+1.5$, $F$ stable on-site soliton at $n=n_0+2$, $G$ unstable intersite soliton at $n=n_0+2.5$, and $H$ stable on-site soliton at $n=n_0+3$. The stable on-site mode located at the impurity, in the homogeneous case, disappears for a coupling value of $C\simeq 1.25$ due to resonances with the phonon band.

Figure 3

Bifurcation loci for different values of parameter $C$. (a) The bifurcation between the on-site localized mode at the impurity $(n=n_0)$ and its neighbor intersite breather $(n=n_0+0.5)$, and (b) the bifurcation between the on-site localized mode next to the impurity $(n=n_0+1)$ and its neighbor intersite breather $(n=n_0+0.5)$. Dashed lines correspond to numerical results and continuous lines to approximate analytical calculations. The curve designation is as follows: $A$ corresponds to $C=1.2$, $B$ to $C=1.1$, $C$ to $C=1.0$, $D$ to $C=0.9$, $E$ to $C=0.8$, and $F$ to $C=0.7$.

Figure 4

Localization as function of amplitude $A$ and impurity parameter $\alpha$ for a single excitation ${\phi }_n(t=0)=A{\delta }_{n,k}$. (a) The excitation localized at the impurity $(k=n_0)$, (b) the excitation localized at the first neighbor of the impurity $(k=n_0+1)$, while (c) is the second neighbor of the impurity $(k=n_0+2)$. The solid line depicts in each case the theoretical threshold given by Eq. (13). In all cases $N=200$ and $C=1.0$.

Figure 5

(a) Power trapping, (b) reflection, and (c) transmission coefficients as function of impurity parameter $\alpha$ and initial thrust $q$. In all cases $N=1000$ and $C=1$.

Figure 6

Trapping: (a) Contour plot corresponding to the power of soliton $P$ as function of site $n$ and time $t$, and (b) Fourier components of the trapped soliton calculated soon after the collision. The parameters are $\alpha =0.2$, $q=0.3$, $\omega =2.5$, $C=1$, and the impurity is located at $n=0$.

Figure 7

Trapping and reflection: (a) Contour plot corresponding to the power of soliton $P$ as a function of site $n$ and time $t$, and (b) Fourier components of the trapped soliton calculated soon after the collision. The parameters are $\alpha =1.0$, $q=0.6$, $\omega =2.5$, $C=1$, and the impurity is located at $n=0$.

Figure 8

(a) Reflection with no trapping corresponding to parameters $\alpha =-0.5$, $q=0.6$, and $\omega =2.5$, and (b) transmission with no trapping corresponding to parameters $\alpha =0.1$, $q=0.7$, and $\omega =2.5$. In both cases we represent a contour plot corresponding to the power of soliton $P$ as function of site $n$ and time $t$, $C=1$, and the impurity is located at $n=0$.

Figure 9

(a) Contour plot of the phenomenon of reflection of a soliton corresponding to trust parameter $q=0.6$. (b) Potential barrier calculated as described in text. In both cases $\alpha =-0.2$, $C=1$, $\omega =2.5$, and the impurity is located at $n=0$.

Figure 10

First winding of the homoclinic tangle of the map (A1). Dashed line corresponds to the linear transformed unstable manifold when $\alpha =0$. Labels $1$, $2$, $3$ ($1^{\prime }$, $2^{\prime }$, $3^{\prime }$) corresponds to fundamental solitons for $\alpha =0$ $(\alpha \ne 0)$.