Stationary scattering from a nonlinear network

Abstract

Transmission through a complex network of nonlinear one-dimensional leads is discussed by extending the stationary scattering theory on quantum graphs to the nonlinear regime. We show that the existence of cycles inside the graph leads to a large number of sharp resonances that dominate scattering. The latter resonances are then shown to be extremely sensitive to the nonlinearity and display multi-stability and hysteresis. This work provides a framework for the study of light propagation in complex optical networks.The study of quantum graphs has gained its popularity in recent years [1] not only because graphs emulate successfully complex mesoscopic and optical networks, but also because they manage to reproduce universal properties (such as level statistics, transmission fluctuations and others) observed in generic quantum chaotic systems. Here we generalize quantum graph theory to the nonlinear domain. The theory will be applied in particular to show the effect of nonlinearity on transmission through networks of nonlinear fibers. Our model may also be used as a simple yet non-trivial model where the universal properties derived from detailed numerical computations of Bose-Einstein condensates in non-regular traps [2][3][4][5][6][7] could be further investigated.Scattering is studied as a stationary process. The main finding is that the sharp resonances which dominate scattering in networks with complex connectivity lead to a dramatic amplification of the nonlinear effects: while non-resonant scattering hardly deviates from the predictions of the linear theory, tuning the parameters to a nearby resonance (without changing the incoming field intensity) brings the system into the nonlinear regime which is signaled by multi-stability and hysteresis. For this reason we revisit the theory of scattering in the linear regime and demonstrate that sharp resonances with large amplification of the incoming wave inside the system are very frequent for graphs compared to other complex (chaotic) scattering systems. The origin of this effect can be related to the topology of the graph (existence of cycles) and leads to a power-law distribution for the amplification. I. THE NONLINEAR SCHRÖDINGER EQUATION ON GRAPHSConsider a general metric graph which consists of V vertices connected by B internal bonds and N leads to infinity as illustrated in Fig. 1. The bonds and leads will be collectively referred to as edges. have the coordinate x l ∈ [0, ∞) and x l = 0 is at the vertex where the lead is attached. The wave function on the graph is a bounded piecewise continuous and differentiable function on the edges written collectively as(1)where ψ e (x e ) is the wave functions on the edge e. While the model can describe far more general settings we restrict ourselves in this exploratory work to the discussion of stationary scattering. The wave function on edge e satisfies the stationary nonlinear Schrödinger equationHere, g e is real nonlinear coupling parameter which we assume constant on each edge (but it may take different values ...

Stationary waves on nonlinear quantum graphs. II. Application of canonical perturbation theory in basic graph structures

“…which is consistent with expanding (19) and (20) for small ρ + and solving for k n (N ) including first-order corrections (for either sign of g). Figure 3 compares the exact spectrum and wave functions to the ones obtained using perturbation theory.…”

“…] which is indeed the first-order expansion in ρ + of the exact expression (19). While the shift of the nonlinear wave number and the deformation of the plane wave solution are both affected by the nonlinearity in first-order perturbation theory, we see that even for the simplest graph the two effects enter in different ways, and that some leading nonlinear corrections to spectral curves may be found by only referring to the shift.…”

“…Indeed, it has been observed [19] that nonlinear effects such as multistability in quantum graphs occur generically already when the incoming flow is very low due to a generic mechanism for narrow resonances, the so-called topological resonances [20]. We refer to [20] for a more detailed discussion of topological resonances in linear quantum graphs and to [19] for a numerical analysis how topological resonances magnify nonlinearities and lead to multistability for arbitrarily small incoming flows. Here, we want to describe this mechanism briefly for two example graphs.…”

A solution of nonlinear Schrödinger equation on metric graphs

“…Ref. [11] treated the stationary NLSE in the context of scattering from nonlinear networks. The stationary NLSE with power focusing nonlinearity on star graphs was studied in recent papers [7,8], where the existence of nonlinear stationary states were shown for δ−type boundary conditions.…”

Ground States for NLS on Graphs: a Subtle Interplay of Metric and Topology

“…The interest was initially driven by physical applications (for an exhaustive introduction see [23]), that involve propagation in optical fibers and junctions ( [14,18,29]), and Bose-Einstein condensation ( [26,28,30]). Subsequently, interesting mathematical issues also arose and have been considered as relevant for themselves.…”