Transport in simple networks described by integrable discrete nonlinear Schr\"Aodinger equation

TL;DR

This paper demonstrates that the Ablowitz-Ladik discrete nonlinear Schrödinger equation on simple networks remains integrable, with solutions related to 1D AL solitons, under specific nonlinearity conditions, leading to conserved quantities and predictable transmission.

Contribution

It extends the integrability of the AL discrete NLSE from 1D chains to simple network structures with bond-dependent nonlinearities, identifying conditions for solutions and conservation laws.

Findings

Solutions are bond-dependent AL solitons scaled by nonlinearity

Nonlinearity sum rule at network vertices ensures integrability

Transmission probabilities are inversely proportional to nonlinearity strength

Abstract

We elucidate the case in which the Ablowitz-Ladik (AL) type discrete nonlinear Schr\"Aodinger equa- tion (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.