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The explosive growth of broadband applications calls for large-scale optical cross-connects (OXCs). However, the classical wavelength selective switch (WSS) based OXC is not scalable in terms of the size of employed WSSs and the cabling complexity. To solve this problem, we propose a three-phase approach to construct a modular WSS-based OXC. In phase 1, we factorize the interconnection network between the input stage and the output stage of the traditional OXC into a set of small-size interconnection networks. In phase 2, we decompose each WSS into a two-stage cascaded structure of small-size WSSs. In phase 3, we combine the small-size interconnection networks with the small-size WSSs to form a set of small-size OXC modules. At last, we obtain a modular OXC, which is a network of small-size OXCs. Similar to the classical OXC, the modular OXC is nonblocking at each wavelength and possesses a self-routing property. Our analysis shows that the modular OXC has small cabling complexity and acceptable physical-layer performance.

In this paper squared eigenfunction symmetry of the differential-difference modified Kadomtsev-Petviashvili (D$\Delta$mKP) hierarchy and its constraint are considered. Under the constraint, the Lax triplets of the D$\Delta$mKP hierarchy, together with their adjoint forms, give rise to the positive relativistic Toda (R-Toda) hierarchy. An invertible transformation is given to connect the positive and negative R-Toda hierarchies. The positive R-Toda hierarchy is reduced to the differential-difference Burgers hierarchy. We also consider another D$\Delta$mKP hierarchy and show that its squared eigenfunction symmetry constraint gives rise to the Volterra hierarchy. In addition, we revisit the Ragnisco-Tu hierarchy which is a squared eigenfunction symmetry constraint of the differential-difference Kadomtsev-Petviashvili (D$\Delta$KP) system. It was thought the Ragnisco-Tu hierarchy does not exist one-field reduction, but here we find an one-field reduction to reduce the hierarchy to the Volterra hierarchy. Besides, the differential-difference Burgers hierarchy are also investigated in Appendix. A multi-dimensionally consistent 3-point discrete Burgers equation is given., Comment: 32 pages, with an Appendix

Let $G$ be a locally compact group, $w$ be a weight on $G$ and $\Phi$ be a Young function. We give some characterizations for translation operators to be topologically transitive and chaotic on the weighted Orlicz space $L_w^\Phi(G)$. In particular, transitivity is equivalent to the blow-up/collapse property in our case. Moreover, the dense set of periodic elements implies transitivity automatically.

In this paper, we study distributional chaos for weighted translations on locally compact groups. We give a sufficient condition for such operators to be distributionally chaotic and construct an example of distributionally chaotic weighted translations by way of the sufficient condition. In particular, we prove the existence of distributional chaos and Li-Yorke chaos for weighted translations operators with aperiodic elements. Furthermore, we also investigate the set of distributionally irregular vectors ($DIV$) of weighted translations through the cone and equivalence classes. When the field is that of complex numbers, we uncover several properties on certain subsets of $DIV$, including their connectedness and correspondences with some measurable subsets in locally compact groups.

The paper presents hodograph transformation between nonlocal short pulse models and the first member in the AKNS negative hierarchy (AKNS($-1$)). We consider real and complex multi-component cases. It is shown that the independent variables of the short pulse models and AKNS($-1$) that are connected via hodograph transformation are covariant in nonlocal reductions.

We use the generization of Weyl's equidistribution theorem to characterize several necessary conditions of hypercyclic weighted translation operators with periodic element., Comment: 14 pages

In the paper possible local and nonlocal reductions of the Ablowitz-Kaup-Newell-Suger (AKNS) hierarchy are collected, including the Korteweg-de Vries (KdV) hierarchy, modified KdV hierarchy and their nonlocal versions, nonlinear Schr\"{o}dinger hierarchy and their nonlocal versions, sine-Gordon equation in nonpotential form and its nonlocal forms. A reduction technique for solutions is employed, by which exact solutions in double Wronskian form are obtained for these reduced equations from those double Wronskian solutions of the AKNS hierarchy. As examples of dynamics we illustrate new interaction of two-soliton solutions of the reverse-$t$ nonlinear Schr\"{o}dinger equation. Although as a single soliton it is always stationary, two solitons travel along completely symmetric trajectories in $\{x,t\}$ plane and their amplitudes are affected by phase parameters. Asymptotic analysis is given as demonstration. The approach and relation described in this paper are systematic and general and can be used to other nonlocal equations., Comment: 26 pages, 4 figures