Transport in simple networks described by an integrable discrete nonlinear Schrödinger equation

Nonlocal nonlinear Schrödinger equation on metric graphs: A model for generation and transport of parity-time-symmetric nonlocal solitons in networks

We consider the parity-time (PT)-symmetric, nonlocal, nonlinear Schrödinger equation on metric graphs. Vertex boundary conditions are derived from the conservation laws. Soliton solutions are obtained for the simplest graph topologies, such as star and tree graphs. The integrability of the problem is shown by proving the existence of an infinite number of conservation laws. A model for soliton generation in such PT-symmetric optical fibers and their networks governed by the nonlocal nonlinear Schrödinger equation is proposed. Exact formulas for the number of generated solitons are derived for the cases when the problem is integrable. Numerical solutions are obtained for the case when integrability is broken.

Transparent nonlinear networks

We consider the reflectionless transport of solitons in networks. The system is modeled in terms of the nonlinear Schrödinger equation on metric graphs, for which transparent boundary conditions at the branching points are imposed. This approach allows to derive simple constraints, which link the equivalent usual Kirchhoff-type vertex conditions to the transparent ones. Our method is applied to a metric star graph. An extension to more complicated graph topologies is straightforward.