Topology-driven nonlinear switching in Möbius discrete arrays

We examine the switching dynamics of discrete solitons propagating along two coupled discrete arrays which are twisted to form a Möbius strip. We analyze the potential of the topological switches by comparing the differences between the Möbius strip and untwisted discrete arrays. We employ the Ablowitz-Ladik (AL) model and reveal a nontrivial Berry phase associated with the monopole spectra in parameter space. We study the dynamical evolution of the AL soliton launched into one of the chains and observe its switching behavior. While in the untwisted discrete case, the soliton splits in nearly identical portions as the interchain coupling is increased, in the Möbius case and for weak coupling, we observe a well-defined “switching time” where the soliton switches completely from one chain to the other.

Figure 1

Geometry of the two ribbons considered in this work: (a) Mobius and (b) untwisted.

Figure 2

Untwisted case. Soliton norm of the AL soliton on chain $A$ (left column) and on chain $B$ (right column), for two values of interchain coupling $\left(\alpha =1,\beta =0.5,N=103\right)$.

Figure 3

Untwisted case. Time evolution of $u_n\left(z\right)$ (left) and $v_n\left(z\right)$ (right) for an initial AL soliton for a normal untwisted strip, and for a small transverse coupling value $t^{\prime }=0.2$ $\left(\alpha =1,\beta =0.5,N=103\right)$.

Figure 4

Untwisted case. Time evolution of $u_n\left(z\right)$ (left) and $v_n\left(z\right)$ (right) for an initial AL soliton for a normal untwisted strip, and for a transverse coupling value $t^{\prime }=0.5$ $\left(\alpha =1,\beta =0.5,N=103\right)$.

Figure 5

Möbius case. Soliton norm of the AL soliton in chains $A$ (left) and $B$ (right) as a function of time, for interchain coupling $t^{\prime }=0$ and $t^{\prime }=0.2$ $\left(\alpha =1,\beta =0.5,N=433\right)$.

Figure 6

Möbius case. Time evolution of $u_n\left(z\right)$ (left) and $v_n\left(z\right)$ (right) for an initial AL soliton on a Möbius strip, in the absence of transversal coupling $t^{\prime }=0$ $\left(\alpha =1,\beta =0.5,N=103\right)$.

Figure 7

Möbius case. Time evolution of $u_n\left(z\right)$ (left) and $v_n\left(z\right)$ (right) for an initial AL soliton on a Möbius strip, for transversal coupling $t^{\prime }=0.26$ $\left(\alpha =1,\beta =0.5,N=103\right)$.

Figure 8

Möbius case. Time evolution of $u_n\left(z\right)$ (left) and $v_n\left(z\right)$ (right) for an initial AL soliton on a Möbius strip, for transversal coupling $t^{\prime }=0.5$ $\left(\alpha =1,\beta =0.5,N=103\right)$.