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

Abstract

We elucidate the case in which the Ablowitz-Ladik (AL) type discrete nonlinear Schrödinger equation (NLSE) on simple networks (e.g., star graphs and tree graphs) becomes completely integrable just as in the case of a simple 1-dimensional (1-d) discrete chain. The strength of cubic nonlinearity is different from bond to bond, and networks are assumed to have at least two semi-infinite bonds with one of them working as an incoming bond. The present work is a nontrivial extension of our preceding one (Sobirov et al, Phys. Rev. E 81, 066602 (2010)) on the continuum NLSE to the discrete case. We find: (1) the solution on each bond is a part of the universal (bond-independent) AL soliton solution on the 1-d discrete chain, but is multiplied by the inverse of square root of bond-dependent nonlinearity; (2) nonlinearities at individual bonds around each vertex must satisfy a sum rule; (3) under findings (1) and (2), there exist an infinite number of constants of motion. As a practical issue, with use of AL soliton injected through the incoming bond, we obtain transmission probabilities inversely proportional to the strength of nonlinearity on the outgoing bonds.

Dispersive effects and high frequency behaviour for the Schrödinger equation in star-shaped networks

“…The results of this article could serve to generalize our results to star shaped networks with general transmission conditions. The article [21] considers discrete analogs of nonlinear Schrödinger equations on star-shaped networks including the existence of solitons, constants of motion and the calculus of transmission probabilities.…”

Sine-Gordon solitons in networks: Scattering and transmission at vertices

“…Early studies of nonlinear evolution equations in branched structures are [9][10][11], and in recent few years one can observe rapidly growing interest in nonlinear waves and soliton transport in networks described by nonlinear Schrödinger equation [12][13][14][15][16][17]. Integrable boundary conditions following from the conservation laws were formulated, and soliton solutions yielding reflectionless transport across the graph vertex were derived in [12], see also [18] for the case of a discrete nonlinear Schrödinger equation. Burioni et al [19,20] studied the discrete nonlinear Schrödinger equation and computed transport and reflection coefficients as a function of the wavenumber of a Gaussian wave packet and the length of a graph attached to a defect site.…”

The stationary sine-Gordon equation on metric graphs: Exact analytical solutions for simple topologies

“…Nonlinear wave equations have found numerous applications in different topics of physics and natural sciences (see, e.g., [1][2][3][4][5][6]). Recently they have attracted much attention in the context of soliton transport in networks and branched structures [7][8][9][10][11][12][13][14][15][16][17][18]. Wave dynamics in networks can be modeled by nonlinear evolution equations on metric graphs.…”